  
  [1X8 [33X[0;0YPermutation Characters in [5XGAP[105X[101X[1X[133X[101X
  
  [33X[0;0YDate: April 17th, 1999[133X
  
  [33X[0;0YThis  is  a  loose  collection  of examples of computations with permutation
  characters and possible permutation characters in the [5XGAP[105X system [GAP21]. We
  mainly  use  the  [5XGAP[105X  implementation  of the algorithms to compute possible
  permutation  characters  that  are described in [BP98], and information from
  the  Atlas  of Finite Groups [CCN+85]. A [13Xpossible permutation character[113X of a
  finite  group  [22XG[122X  is a character satisfying the conditions listed in Section
  [21XPossible Permutation Characters[121X of the [5XGAP[105X Reference Manual.[133X
  
  [30X    [33X[0;6YSections [14X8.14[114X and [14X8.15[114X were added in October 2001.[133X
  
  [30X    [33X[0;6YSection [14X8.16-1[114X was added in June 2009.[133X
  
  [30X    [33X[0;6YSection [14X8.16-2[114X was added in September 2009.[133X
  
  [30X    [33X[0;6YSection [14X8.16-3[114X was added in October 2009.[133X
  
  [30X    [33X[0;6YSection [14X8.16-4[114X was added in November 2009.[133X
  
  [30X    [33X[0;6YSection [14X8.17[114X was added in June 2012.[133X
  
  [30X    [33X[0;6YSection [14X8.18[114X was added in October 2017.[133X
  
  [30X    [33X[0;6YSection [14X8.19[114X was added in December 2021.[133X
  
  [33X[0;0YIn  the  following,  the  [5XGAP[105X  Character  Table Library [Bre22] will be used
  frequently.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XLoadPackage( "ctbllib", "1.2", false );[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  
  [1X8.1 [33X[0;0YSome Computations with [22XM_24[122X[101X[1X[133X[101X
  
  [33X[0;0YWe  start  with  the  sporadic  simple Mathieu group [22XG = M_24[122X in its natural
  action on [22X24[122X points.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xg:= MathieuGroup( 24 );;[127X[104X
    [4X[25Xgap>[125X [27XSetName( g, "m24" );[127X[104X
    [4X[25Xgap>[125X [27XSize( g );  IsSimple( g );  NrMovedPoints( g );[127X[104X
    [4X[28X244823040[128X[104X
    [4X[28Xtrue[128X[104X
    [4X[28X24[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  conjugacy  classes  that  are  computed  for  a  group  can  be ordered
  differently  in different [5XGAP[105X sessions. In order to make the output shown in
  the  following examples stable, we first sort the conjugacy classes of [22XG[122X for
  our purposes.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xccl:= AttributeValueNotSet( ConjugacyClasses, g );;[127X[104X
    [4X[25Xgap>[125X [27XHasConjugacyClasses( g );[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27Xinvariants:= List( ccl, c -> [ Order( Representative( c ) ),[127X[104X
    [4X[25X>[125X [27X       Size( c ), Size( ConjugacyClass( g, Representative( c )^2 ) ) ] );;[127X[104X
    [4X[25Xgap>[125X [27XSortParallel( invariants, ccl );[127X[104X
    [4X[25Xgap>[125X [27XSetConjugacyClasses( g, ccl );[127X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  permutation  character [10Xpi[110X of [22XG[122X corresponding to the action on the moved
  points is constructed. This action is [22X5[122X-transitive.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XNrConjugacyClasses( g );[127X[104X
    [4X[28X26[128X[104X
    [4X[25Xgap>[125X [27Xpi:= NaturalCharacter( g );[127X[104X
    [4X[28XCharacter( CharacterTable( m24 ),[128X[104X
    [4X[28X [ 24, 8, 0, 6, 0, 0, 4, 0, 4, 2, 0, 3, 3, 2, 0, 2, 0, 0, 1, 1, 1, 1, [128X[104X
    [4X[28X  0, 0, 1, 1 ] )[128X[104X
    [4X[25Xgap>[125X [27XIsTransitive( pi );  Transitivity( pi );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[28X5[128X[104X
    [4X[25Xgap>[125X [27XDisplay( pi );[127X[104X
    [4X[28XCT1[128X[104X
    [4X[28X[128X[104X
    [4X[28X     2 10 10  9  3  3  7  7  5  2  3  3  1  1  4   2   .   2   2   1[128X[104X
    [4X[28X     3  3  1  1  3  2  1  .  1  1  1  1  1  1  .   .   .   1   1   .[128X[104X
    [4X[28X     5  1  .  1  1  .  .  .  .  1  .  .  .  .  .   1   .   .   .   .[128X[104X
    [4X[28X     7  1  1  .  .  1  .  .  .  .  .  .  1  1  .   .   .   .   .   1[128X[104X
    [4X[28X    11  1  .  .  .  .  .  .  .  .  .  .  .  .  .   .   1   .   .   .[128X[104X
    [4X[28X    23  1  .  .  .  .  .  .  .  .  .  .  .  .  .   .   .   .   .   .[128X[104X
    [4X[28X[128X[104X
    [4X[28X       1a 2a 2b 3a 3b 4a 4b 4c 5a 6a 6b 7a 7b 8a 10a 11a 12a 12b 14a[128X[104X
    [4X[28X[128X[104X
    [4X[28XY.1    24  8  .  6  .  .  4  .  4  2  .  3  3  2   .   2   .   .   1[128X[104X
    [4X[28X[128X[104X
    [4X[28X     2   1   .   .   .   .   .   .[128X[104X
    [4X[28X     3   .   1   1   1   1   .   .[128X[104X
    [4X[28X     5   .   1   1   .   .   .   .[128X[104X
    [4X[28X     7   1   .   .   1   1   .   .[128X[104X
    [4X[28X    11   .   .   .   .   .   .   .[128X[104X
    [4X[28X    23   .   .   .   .   .   1   1[128X[104X
    [4X[28X[128X[104X
    [4X[28X       14b 15a 15b 21a 21b 23a 23b[128X[104X
    [4X[28X[128X[104X
    [4X[28XY.1      1   1   1   .   .   1   1[128X[104X
  [4X[32X[104X
  
  [33X[0;0Y[10Xpi[110X  determines  the permutation characters of the [22XG[122X-actions on related sets,
  for  example  [10Xpiop[110X  on  the  set of ordered and [10Xpiup[110X on the set of unordered
  pairs of points.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xpiop:= pi * pi;[127X[104X
    [4X[28XCharacter( CharacterTable( m24 ),[128X[104X
    [4X[28X [ 576, 64, 0, 36, 0, 0, 16, 0, 16, 4, 0, 9, 9, 4, 0, 4, 0, 0, 1, 1, [128X[104X
    [4X[28X  1, 1, 0, 0, 1, 1 ] )[128X[104X
    [4X[25Xgap>[125X [27XIsTransitive( piop );[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27Xpiup:= SymmetricParts( UnderlyingCharacterTable(pi), [ pi ], 2 )[1];[127X[104X
    [4X[28XCharacter( CharacterTable( m24 ),[128X[104X
    [4X[28X [ 300, 44, 12, 21, 0, 4, 12, 0, 10, 5, 0, 6, 6, 4, 2, 3, 1, 0, 2, 2, [128X[104X
    [4X[28X  1, 1, 0, 0, 1, 1 ] )[128X[104X
    [4X[25Xgap>[125X [27XIsTransitive( piup );[127X[104X
    [4X[28Xfalse[128X[104X
  [4X[32X[104X
  
  [33X[0;0YClearly  the  action on unordered pairs is not transitive, since the pairs [22X[
  i,  i  ][122X  form  an orbit of their own. There are exactly two [22XG[122X-orbits on the
  unordered pairs, hence the [22XG[122X-action on [22X2[122X-sets of points is transitive.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XScalarProduct( piup, TrivialCharacter( g ) );[127X[104X
    [4X[28X2[128X[104X
    [4X[25Xgap>[125X [27Xcomb:= Combinations( [ 1 .. 24 ], 2 );;[127X[104X
    [4X[25Xgap>[125X [27Xhom:= ActionHomomorphism( g, comb, OnSets );;[127X[104X
    [4X[25Xgap>[125X [27Xpihom:= NaturalCharacter( hom );[127X[104X
    [4X[28XCharacter( CharacterTable( m24 ),[128X[104X
    [4X[28X [ 276, 36, 12, 15, 0, 4, 8, 0, 6, 3, 0, 3, 3, 2, 2, 1, 1, 0, 1, 1, [128X[104X
    [4X[28X  0, 0, 0, 0, 0, 0 ] )[128X[104X
    [4X[25Xgap>[125X [27XTransitivity( pihom );[127X[104X
    [4X[28X1[128X[104X
  [4X[32X[104X
  
  [33X[0;0YIn terms of characters, the permutation character [10Xpihom[110X is the difference of
  [10Xpiup[110X  and [10Xpi[110X . Note that [5XGAP[105X does not know that this difference is in fact a
  character;  in  general  this question is not easy to decide without knowing
  the  irreducible  characters  of  [22XG[122X,  and up to now [5XGAP[105X has not computed the
  irreducibles.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xpi2s:= piup - pi;[127X[104X
    [4X[28XVirtualCharacter( CharacterTable( m24 ),[128X[104X
    [4X[28X [ 276, 36, 12, 15, 0, 4, 8, 0, 6, 3, 0, 3, 3, 2, 2, 1, 1, 0, 1, 1, [128X[104X
    [4X[28X  0, 0, 0, 0, 0, 0 ] )[128X[104X
    [4X[25Xgap>[125X [27Xpi2s = pihom;[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XHasIrr( g );  HasIrr( CharacterTable( g ) );[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[28Xfalse[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  point  stabilizer in the action on [22X2[122X-sets is in fact a maximal subgroup
  of  [22XG[122X,  which  is isomorphic to the automorphism group [22XM_22:2[122X of the Mathieu
  group  [22XM_22[122X.  Thus this permutation action is primitive. But we cannot apply
  [2XIsPrimitive[102X ([14XReference: IsPrimitive[114X) to the character [10Xpihom[110X for getting this
  answer  because  primitivity  of  characters  is defined in a different way,
  cf. [2XIsPrimitiveCharacter[102X ([14XReference: IsPrimitiveCharacter[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XIsPrimitive( g, comb, OnSets );[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [33X[0;0YWe  could  also have computed the transitive permutation character of degree
  [22X276[122X  using the [5XGAP[105X Character Table Library instead of the group [22XG[122X, since the
  character  tables of [22XG[122X and all its maximal subgroups are available, together
  with the class fusions of the maximal subgroups into [22XG[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xtbl:= CharacterTable( "M24" );[127X[104X
    [4X[28XCharacterTable( "M24" )[128X[104X
    [4X[25Xgap>[125X [27Xmaxes:= Maxes( tbl );[127X[104X
    [4X[28X[ "M23", "M22.2", "2^4:a8", "M12.2", "2^6:3.s6", "L3(4).3.2_2", [128X[104X
    [4X[28X  "2^6:(psl(3,2)xs3)", "L2(23)", "L3(2)" ][128X[104X
    [4X[25Xgap>[125X [27Xs:= CharacterTable( maxes[2] );[127X[104X
    [4X[28XCharacterTable( "M22.2" )[128X[104X
    [4X[25Xgap>[125X [27XTrivialCharacter( s )^tbl;[127X[104X
    [4X[28XCharacter( CharacterTable( "M24" ),[128X[104X
    [4X[28X [ 276, 36, 12, 15, 0, 4, 8, 0, 6, 3, 0, 3, 3, 2, 2, 1, 1, 0, 1, 1, [128X[104X
    [4X[28X  0, 0, 0, 0, 0, 0 ] )[128X[104X
  [4X[32X[104X
  
  [33X[0;0YNote  that  the sequence of conjugacy classes in the library table of [22XG[122X does
  in general not agree with the succession computed for the group.[133X
  
  
  [1X8.2 [33X[0;0YAll Possible Permutation Characters of [22XM_11[122X[101X[1X[133X[101X
  
  [33X[0;0YWe  compute  all  possible permutation characters of the Mathieu group [22XM_11[122X,
  using  the  three  different  strategies  available in [5XGAP[105X. First we try the
  algorithm   that   enumerates   all  candidates  via  solving  a  system  of
  inequalities, which is described in [BP98, Section 3.2].[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xm11:= CharacterTable( "M11" );;[127X[104X
    [4X[25Xgap>[125X [27XSetName( m11, "m11" );[127X[104X
    [4X[25Xgap>[125X [27Xperms:= PermChars( m11 );[127X[104X
    [4X[28X[ Character( m11, [ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ] ), Character( m11,[128X[104X
    [4X[28X  [ 11, 3, 2, 3, 1, 0, 1, 1, 0, 0 ] ), Character( m11,[128X[104X
    [4X[28X  [ 12, 4, 3, 0, 2, 1, 0, 0, 1, 1 ] ), Character( m11,[128X[104X
    [4X[28X  [ 22, 6, 4, 2, 2, 0, 0, 0, 0, 0 ] ), Character( m11,[128X[104X
    [4X[28X  [ 55, 7, 1, 3, 0, 1, 1, 1, 0, 0 ] ), Character( m11,[128X[104X
    [4X[28X  [ 66, 10, 3, 2, 1, 1, 0, 0, 0, 0 ] ), Character( m11,[128X[104X
    [4X[28X  [ 110, 6, 2, 2, 0, 0, 2, 2, 0, 0 ] ), Character( m11,[128X[104X
    [4X[28X  [ 110, 6, 2, 6, 0, 0, 0, 0, 0, 0 ] ), Character( m11,[128X[104X
    [4X[28X  [ 110, 14, 2, 2, 0, 2, 0, 0, 0, 0 ] ), Character( m11,[128X[104X
    [4X[28X  [ 132, 12, 6, 0, 2, 0, 0, 0, 0, 0 ] ), Character( m11,[128X[104X
    [4X[28X  [ 144, 0, 0, 0, 4, 0, 0, 0, 1, 1 ] ), Character( m11,[128X[104X
    [4X[28X  [ 165, 13, 3, 1, 0, 1, 1, 1, 0, 0 ] ), Character( m11,[128X[104X
    [4X[28X  [ 220, 4, 4, 0, 0, 4, 0, 0, 0, 0 ] ), Character( m11,[128X[104X
    [4X[28X  [ 220, 12, 4, 4, 0, 0, 0, 0, 0, 0 ] ), Character( m11,[128X[104X
    [4X[28X  [ 220, 20, 4, 0, 0, 2, 0, 0, 0, 0 ] ), Character( m11,[128X[104X
    [4X[28X  [ 330, 2, 6, 2, 0, 2, 0, 0, 0, 0 ] ), Character( m11,[128X[104X
    [4X[28X  [ 330, 18, 6, 2, 0, 0, 0, 0, 0, 0 ] ), Character( m11,[128X[104X
    [4X[28X  [ 396, 12, 0, 4, 1, 0, 0, 0, 0, 0 ] ), Character( m11,[128X[104X
    [4X[28X  [ 440, 8, 8, 0, 0, 2, 0, 0, 0, 0 ] ), Character( m11,[128X[104X
    [4X[28X  [ 440, 24, 8, 0, 0, 0, 0, 0, 0, 0 ] ), Character( m11,[128X[104X
    [4X[28X  [ 495, 15, 0, 3, 0, 0, 1, 1, 0, 0 ] ), Character( m11,[128X[104X
    [4X[28X  [ 660, 4, 3, 4, 0, 1, 0, 0, 0, 0 ] ), Character( m11,[128X[104X
    [4X[28X  [ 660, 12, 3, 0, 0, 3, 0, 0, 0, 0 ] ), Character( m11,[128X[104X
    [4X[28X  [ 660, 12, 12, 0, 0, 0, 0, 0, 0, 0 ] ), Character( m11,[128X[104X
    [4X[28X  [ 660, 28, 3, 0, 0, 1, 0, 0, 0, 0 ] ), Character( m11,[128X[104X
    [4X[28X  [ 720, 0, 0, 0, 0, 0, 0, 0, 5, 5 ] ), Character( m11,[128X[104X
    [4X[28X  [ 792, 24, 0, 0, 2, 0, 0, 0, 0, 0 ] ), Character( m11,[128X[104X
    [4X[28X  [ 880, 0, 16, 0, 0, 0, 0, 0, 0, 0 ] ), Character( m11,[128X[104X
    [4X[28X  [ 990, 6, 0, 2, 0, 0, 2, 2, 0, 0 ] ), Character( m11,[128X[104X
    [4X[28X  [ 990, 6, 0, 6, 0, 0, 0, 0, 0, 0 ] ), Character( m11,[128X[104X
    [4X[28X  [ 990, 30, 0, 2, 0, 0, 0, 0, 0, 0 ] ), Character( m11,[128X[104X
    [4X[28X  [ 1320, 8, 6, 0, 0, 2, 0, 0, 0, 0 ] ), Character( m11,[128X[104X
    [4X[28X  [ 1320, 24, 6, 0, 0, 0, 0, 0, 0, 0 ] ), Character( m11,[128X[104X
    [4X[28X  [ 1584, 0, 0, 0, 4, 0, 0, 0, 0, 0 ] ), Character( m11,[128X[104X
    [4X[28X  [ 1980, 12, 0, 4, 0, 0, 0, 0, 0, 0 ] ), Character( m11,[128X[104X
    [4X[28X  [ 1980, 36, 0, 0, 0, 0, 0, 0, 0, 0 ] ), Character( m11,[128X[104X
    [4X[28X  [ 2640, 0, 12, 0, 0, 0, 0, 0, 0, 0 ] ), Character( m11,[128X[104X
    [4X[28X  [ 3960, 24, 0, 0, 0, 0, 0, 0, 0, 0 ] ), Character( m11,[128X[104X
    [4X[28X  [ 7920, 0, 0, 0, 0, 0, 0, 0, 0, 0 ] ) ][128X[104X
    [4X[25Xgap>[125X [27XLength( perms );[127X[104X
    [4X[28X39[128X[104X
  [4X[32X[104X
  
  [33X[0;0YNext  we try the improved combinatorial approach that is sketched at the end
  of  Section 3.2  in [BP98]. We get the same characters, except that they may
  be ordered in a different way; thus we compare the ordered lists.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xdegrees:= DivisorsInt( Size( m11 ) );;[127X[104X
    [4X[25Xgap>[125X [27Xperms2:= [];;[127X[104X
    [4X[25Xgap>[125X [27Xfor d in degrees do[127X[104X
    [4X[25X>[125X [27X     Append( perms2, PermChars( m11, d ) );[127X[104X
    [4X[25X>[125X [27X   od;[127X[104X
    [4X[25Xgap>[125X [27XSet( perms ) = Set( perms2 );[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [33X[0;0YFinally, we try the algorithm that is based on Gaussian elimination and that
  is described in [BP98, Section 3.3].[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xperms3:= [];;[127X[104X
    [4X[25Xgap>[125X [27Xfor d in degrees do[127X[104X
    [4X[25X>[125X [27X     Append( perms3, PermChars( m11, rec( torso:= [ d ] ) ) );[127X[104X
    [4X[25X>[125X [27X   od;[127X[104X
    [4X[25Xgap>[125X [27XSet( perms ) = Set( perms3 );[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [33X[0;0Y[5XGAP[105X   provides   two  more  functions  to  test  properties  of  permutation
  characters.  The  first  one  yields no new information in our case, but the
  second  excludes  one  possible  permutation  character; note that [10XTestPerm5[110X
  needs a [22Xp[122X-modular Brauer table, and the [5XGAP[105X character table library contains
  all Brauer tables of [22XM_11[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xnewperms:= TestPerm4( m11, perms );;[127X[104X
    [4X[25Xgap>[125X [27Xnewperms = perms;[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xnewperms:= TestPerm5( m11, perms, m11 mod 11 );;[127X[104X
    [4X[25Xgap>[125X [27Xnewperms = perms;[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XDifference( perms, newperms );[127X[104X
    [4X[28X[ Character( m11, [ 220, 4, 4, 0, 0, 4, 0, 0, 0, 0 ] ) ][128X[104X
  [4X[32X[104X
  
  [33X[0;0Y[5XGAP[105X  knows the table of marks of [22XM_11[122X, from which the permutation characters
  can  be  extracted.  It  turns  out  that  [22XM_11[122X  has [22X39[122X conjugacy classes of
  subgroups  but only [22X36[122X different permutation characters, so three candidates
  computed above are in fact not permutation characters.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xtom:= TableOfMarks( "M11" );[127X[104X
    [4X[28XTableOfMarks( "M11" )[128X[104X
    [4X[25Xgap>[125X [27Xtrueperms:= PermCharsTom( m11, tom );;[127X[104X
    [4X[25Xgap>[125X [27XLength( trueperms );  Length( Set( trueperms ) );[127X[104X
    [4X[28X39[128X[104X
    [4X[28X36[128X[104X
    [4X[25Xgap>[125X [27XDifference( perms, trueperms );[127X[104X
    [4X[28X[ Character( m11, [ 220, 4, 4, 0, 0, 4, 0, 0, 0, 0 ] ), [128X[104X
    [4X[28X  Character( m11, [ 660, 4, 3, 4, 0, 1, 0, 0, 0, 0 ] ), [128X[104X
    [4X[28X  Character( m11, [ 660, 12, 3, 0, 0, 3, 0, 0, 0, 0 ] ) ][128X[104X
  [4X[32X[104X
  
  
  [1X8.3 [33X[0;0YThe Action of [22XU_6(2)[122X[101X[1X on the Cosets of [22XM_22[122X[101X[1X[133X[101X
  
  [33X[0;0YWe  are  interested  in the permutation character of [22XU_6(2)[122X (see [CCN+85, p.
  115])  that  corresponds  to  the  action  on  the cosets of a [22XM_22[122X subgroup
  (see [CCN+85,  p. 39]). The character tables of both the group and the point
  stabilizer  are  available  in  the  [5XGAP[105X  character table library, so we can
  compute  class  fusion  and permutation character directly; note that if the
  class fusion is not stored on the table of the subgroup, in general one will
  not get a unique fusion but only a list of candidates for the fusion.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xu62:= CharacterTable( "U6(2)" );;[127X[104X
    [4X[25Xgap>[125X [27Xm22:= CharacterTable( "M22" );;[127X[104X
    [4X[25Xgap>[125X [27Xfus:= PossibleClassFusions( m22, u62 );[127X[104X
    [4X[28X[ [ 1, 3, 7, 10, 14, 15, 22, 24, 24, 26, 33, 34 ], [128X[104X
    [4X[28X  [ 1, 3, 7, 10, 14, 15, 22, 24, 24, 26, 34, 33 ], [128X[104X
    [4X[28X  [ 1, 3, 7, 11, 14, 15, 22, 24, 24, 27, 33, 34 ], [128X[104X
    [4X[28X  [ 1, 3, 7, 11, 14, 15, 22, 24, 24, 27, 34, 33 ], [128X[104X
    [4X[28X  [ 1, 3, 7, 12, 14, 15, 22, 24, 24, 28, 33, 34 ], [128X[104X
    [4X[28X  [ 1, 3, 7, 12, 14, 15, 22, 24, 24, 28, 34, 33 ] ][128X[104X
    [4X[25Xgap>[125X [27XRepresentativesFusions( m22, fus, u62 );[127X[104X
    [4X[28X[ [ 1, 3, 7, 10, 14, 15, 22, 24, 24, 26, 33, 34 ] ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YWe  see  that there are six possible class fusions that are equivalent under
  table automorphisms of [22XU_6(2)[122X and [22XM22[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xcand:= Set( fus,[127X[104X
    [4X[25X>[125X [27X x -> Induced( m22, u62, [ TrivialCharacter( m22 ) ], x )[1] );[127X[104X
    [4X[28X[ Character( CharacterTable( "U6(2)" ),[128X[104X
    [4X[28X  [ 20736, 0, 384, 0, 0, 0, 54, 0, 0, 0, 0, 48, 0, 16, 6, 0, 0, 0, 0, [128X[104X
    [4X[28X      0, 0, 6, 0, 2, 0, 0, 0, 4, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, [128X[104X
    [4X[28X      0, 0, 0, 0, 0, 0 ] ), Character( CharacterTable( "U6(2)" ),[128X[104X
    [4X[28X  [ 20736, 0, 384, 0, 0, 0, 54, 0, 0, 0, 48, 0, 0, 16, 6, 0, 0, 0, 0, [128X[104X
    [4X[28X      0, 0, 6, 0, 2, 0, 0, 4, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, [128X[104X
    [4X[28X      0, 0, 0, 0, 0, 0 ] ), Character( CharacterTable( "U6(2)" ),[128X[104X
    [4X[28X  [ 20736, 0, 384, 0, 0, 0, 54, 0, 0, 48, 0, 0, 0, 16, 6, 0, 0, 0, 0, [128X[104X
    [4X[28X      0, 0, 6, 0, 2, 0, 4, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, [128X[104X
    [4X[28X      0, 0, 0, 0, 0, 0 ] ) ][128X[104X
    [4X[25Xgap>[125X [27XPermCharInfo( u62, cand ).ATLAS;[127X[104X
    [4X[28X[ "1a+22a+252a+616a+1155c+1386a+8064a+9240c", [128X[104X
    [4X[28X  "1a+22a+252a+616a+1155b+1386a+8064a+9240b", [128X[104X
    [4X[28X  "1a+22a+252a+616a+1155a+1386a+8064a+9240a" ][128X[104X
    [4X[25Xgap>[125X [27Xaut:= AutomorphismsOfTable( u62 );;  Size( aut );[127X[104X
    [4X[28X24[128X[104X
    [4X[25Xgap>[125X [27Xelms:= Filtered( Elements( aut ), x -> Order( x ) = 3 );[127X[104X
    [4X[28X[ (10,11,12)(26,27,28)(40,41,42), (10,12,11)(26,28,27)(40,42,41) ][128X[104X
    [4X[25Xgap>[125X [27XPosition( cand, Permuted( cand[1], elms[1] ) );[127X[104X
    [4X[28X3[128X[104X
    [4X[25Xgap>[125X [27XPosition( cand, Permuted( cand[3], elms[1] ) );[127X[104X
    [4X[28X2[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  six fusions induce three different characters, they are conjugate under
  the  action  of  the  unique  subgroup  of  order  [22X3[122X  in  the group of table
  automorphisms  of  [22XU_6(2)[122X. The table automorphisms of order [22X3[122X are induced by
  group  automorphisms  of  [22XU_6(2)[122X (see [CCN+85, p. 120]). As can be seen from
  the  list  of  maximal  subgroups  of  [22XU_6(2)[122X in [CCN+85, p. 115], the three
  induced  characters  are  in fact permutation characters which belong to the
  three  classes  of  maximal  subgroups  of  type  [22XM_22[122X  in [22XU_6(2)[122X, which are
  permuted  by  an  outer  automorphism of order 3. Now we want to compute the
  extension  of  the  above permutation character to the group [22XU_6(2).2[122X, which
  corresponds to the action of this group on the cosets of a [22XM_22.2[122X subgroup.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xu622:= CharacterTable( "U6(2).2" );;[127X[104X
    [4X[25Xgap>[125X [27Xm222:= CharacterTable( "M22.2" );;[127X[104X
    [4X[25Xgap>[125X [27Xfus:= PossibleClassFusions( m222, u622 );[127X[104X
    [4X[28X[ [ 1, 3, 7, 10, 13, 14, 20, 22, 22, 24, 29, 38, 39, 42, 41, 46, 50, [128X[104X
    [4X[28X      53, 58, 59, 59 ] ][128X[104X
    [4X[25Xgap>[125X [27Xcand:= Induced( m222, u622, [ TrivialCharacter( m222 ) ], fus[1] );[127X[104X
    [4X[28X[ Character( CharacterTable( "U6(2).2" ),[128X[104X
    [4X[28X  [ 20736, 0, 384, 0, 0, 0, 54, 0, 0, 48, 0, 0, 16, 6, 0, 0, 0, 0, 0, [128X[104X
    [4X[28X      6, 0, 2, 0, 4, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1080, 72, [128X[104X
    [4X[28X      0, 48, 8, 0, 0, 0, 18, 0, 0, 0, 8, 0, 0, 2, 0, 0, 0, 0, 2, 2, [128X[104X
    [4X[28X      0, 0, 0, 0, 0, 0 ] ) ][128X[104X
    [4X[25Xgap>[125X [27XPermCharInfo( u622, cand ).ATLAS;[127X[104X
    [4X[28X[ "1a+22a+252a+616a+1155a+1386a+8064a+9240a" ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YWe  see  that for the embedding of [22XM_22.2[122X into [22XU_6(2).2[122X, the class fusion is
  unique,  so  we  get  a  unique  extension  of  one of the above permutation
  characters. This implies that exactly one class of maximal subgroups of type
  [22XM_22[122X extends to [22XM_22.2[122X in a given group [22XU_6(2).2[122X.[133X
  
  
  [1X8.4 [33X[0;0YDegree [22X20736[122X[101X[1X Permutation Characters of [22XU_6(2)[122X[101X[1X[133X[101X
  
  [33X[0;0YNow  we  show an alternative way to compute the characters dealt with in the
  previous  example.  This  works  also  if  the  character table of the point
  stabilizer  is  not  available.  In  this situation we can compute all those
  characters that have certain properties of permutation characters. Of course
  this  may  take much longer than the above computations, which needed only a
  few  seconds.  (The following calculations may need several hours, depending
  on the computer used.)[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xcand:= PermChars( u62, rec( torso := [ 20736 ] ) );[127X[104X
    [4X[28X[ Character( CharacterTable( "U6(2)" ), [128X[104X
    [4X[28X    [ 20736, 0, 384, 0, 0, 0, 54, 0, 0, 0, 0, 48, 0, 16, 6, 0, 0, 0, [128X[104X
    [4X[28X      0, 0, 0, 6, 0, 2, 0, 0, 0, 4, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, [128X[104X
    [4X[28X      0, 0, 0, 0, 0, 0, 0 ] ), Character( CharacterTable( "U6(2)" ), [128X[104X
    [4X[28X    [ 20736, 0, 384, 0, 0, 0, 54, 0, 0, 0, 48, 0, 0, 16, 6, 0, 0, 0, [128X[104X
    [4X[28X      0, 0, 0, 6, 0, 2, 0, 0, 4, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, [128X[104X
    [4X[28X      0, 0, 0, 0, 0, 0, 0 ] ), Character( CharacterTable( "U6(2)" ), [128X[104X
    [4X[28X    [ 20736, 0, 384, 0, 0, 0, 54, 0, 0, 48, 0, 0, 0, 16, 6, 0, 0, 0, [128X[104X
    [4X[28X      0, 0, 0, 6, 0, 2, 0, 4, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, [128X[104X
    [4X[28X      0, 0, 0, 0, 0, 0, 0 ] ) ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YFor  the  next  step,  that  is,  the  computation  of  the extension of the
  permutation  character  to [22XU_6(2).2[122X, we may use the above information, since
  the  values  on  the inner classes are prescribed. The question which of the
  three candidates for [22XU_6(2)[122X extends to [22XU_6(2).2[122X depends on the choice of the
  class  fusion of [22XU_6(2)[122X into [22XU_6(2).2[122X. With respect to the class fusion that
  is  stored  on the [5XGAP[105X library table, the third candidate extends, as can be
  seen  from  the  fact  that  this  one is invariant under the permutation of
  conjugacy  classes  of  [22XU_6(2)[122X  that  is induced by the action of the chosen
  supergroup [22XU_6(2).2[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xu622:= CharacterTable( "U6(2).2" );;[127X[104X
    [4X[25Xgap>[125X [27Xinv:= InverseMap( GetFusionMap( u62, u622 ) );[127X[104X
    [4X[28X[ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, [ 11, 12 ], 13, 14, 15, [ 16, 17 ], [128X[104X
    [4X[28X  18, 19, 20, 21, 22, 23, 24, 25, 26, [ 27, 28 ], [ 29, 30 ], 31, 32, [128X[104X
    [4X[28X  [ 33, 34 ], [ 35, 36 ], 37, [ 38, 39 ], 40, [ 41, 42 ], 43, 44, [128X[104X
    [4X[28X  [ 45, 46 ] ][128X[104X
    [4X[25Xgap>[125X [27Xext:= List( cand, x -> CompositionMaps( x, inv ) );[127X[104X
    [4X[28X[ [ 20736, 0, 384, 0, 0, 0, 54, 0, 0, 0, [ 0, 48 ], 0, 16, 6, 0, 0, [128X[104X
    [4X[28X      0, 0, 0, 6, 0, 2, 0, 0, [ 0, 4 ], 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, [128X[104X
    [4X[28X      0, 0 ], [128X[104X
    [4X[28X  [ 20736, 0, 384, 0, 0, 0, 54, 0, 0, 0, [ 0, 48 ], 0, 16, 6, 0, 0, [128X[104X
    [4X[28X      0, 0, 0, 6, 0, 2, 0, 0, [ 0, 4 ], 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, [128X[104X
    [4X[28X      0, 0 ], [128X[104X
    [4X[28X  [ 20736, 0, 384, 0, 0, 0, 54, 0, 0, 48, 0, 0, 16, 6, 0, 0, 0, 0, 0, [128X[104X
    [4X[28X      6, 0, 2, 0, 4, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0 ] ][128X[104X
    [4X[25Xgap>[125X [27Xcand:= PermChars( u622, rec( torso:= ext[3] ) );[127X[104X
    [4X[28X[ Character( CharacterTable( "U6(2).2" ), [128X[104X
    [4X[28X    [ 20736, 0, 384, 0, 0, 0, 54, 0, 0, 48, 0, 0, 16, 6, 0, 0, 0, 0, [128X[104X
    [4X[28X      0, 6, 0, 2, 0, 4, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1080, [128X[104X
    [4X[28X      72, 0, 48, 8, 0, 0, 0, 18, 0, 0, 0, 8, 0, 0, 2, 0, 0, 0, 0, 2, [128X[104X
    [4X[28X      2, 0, 0, 0, 0, 0, 0 ] ) ][128X[104X
  [4X[32X[104X
  
  
  [1X8.5 [33X[0;0YDegree [22X57572775[122X[101X[1X Permutation Characters of [22XO_8^+(3)[122X[101X[1X[133X[101X
  
  [33X[0;0YThe  group  [22XO_8^+(3)[122X  (see [CCN+85,  p.  140])  contains  a subgroup of type
  [22X2^{3+6}.L_3(2)[122X, which extends to a maximal subgroup [22XU[122X in [22XO_8^+(3).3[122X. For the
  computation  of  the permutation character, we cannot use explicit induction
  since  the  table  of [22XU[122X is not available in the [5XGAP[105X table library. Since [22XU ∩
  O_8^+(3)[122X is contained in a [22XO_8^+(2)[122X subgroup of [22XO_8^+(3)[122X, we can try to find
  the  permutation  character  of  [22XO_8^+(2)[122X corresponding to the action on the
  cosets  of  [22XU  ∩  O_8^+(3)[122X, and then induce this character to [22XO_8^+(3)[122X. This
  kind  of  computations  becomes more difficult with increasing degree, so we
  try  to  reduce  the  problem  further. In fact, the [22X2^{3+6}.L_3(2)[122X group is
  contained  in a [22X2^6:A_8[122X subgroup of [22XO_8^+(2)[122X, in which the index is only [22X15[122X;
  the  unique  possible  permutation  character of this degree can be read off
  immediately.  Induction  to  [22XO_8^+(3)[122X  through  the  chain  of  subgroups is
  possible  provided  the  class  fusions are available. There are [22X24[122X possible
  fusions  from  [22XO_8^+(2)[122X into [22XO_8^+(3)[122X, which are all equivalent w.r.t. table
  automorphisms of [22XO_8^+(3)[122X. If we later want to consider the extension of the
  permutation  character  in  question  to [22XO_8^+(3).3[122X then we have to choose a
  fusion  of  an  [22XO_8^+(2)[122X subgroup that does [13Xnot[113X extend to [22XO_8^+(2).3[122X. But if
  for example our question is just whether the resulting permutation character
  is  multiplicity-free  then this can be decided already from the permutation
  character of [22XO_8^+(3)[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xo8p3:= CharacterTable("O8+(3)");;[127X[104X
    [4X[25Xgap>[125X [27XSize( o8p3 ) / (2^9*168);[127X[104X
    [4X[28X57572775[128X[104X
    [4X[25Xgap>[125X [27Xo8p2:= CharacterTable( "O8+(2)" );;[127X[104X
    [4X[25Xgap>[125X [27Xfus:= PossibleClassFusions( o8p2, o8p3 );;[127X[104X
    [4X[25Xgap>[125X [27XLength( fus );[127X[104X
    [4X[28X24[128X[104X
    [4X[25Xgap>[125X [27Xrep:= RepresentativesFusions( o8p2, fus, o8p3 );[127X[104X
    [4X[28X[ [ 1, 5, 2, 3, 4, 5, 7, 8, 12, 16, 17, 19, 23, 20, 21, 22, 23, 24, [128X[104X
    [4X[28X      25, 26, 37, 38, 42, 31, 32, 36, 49, 52, 51, 50, 43, 44, 45, 53, [128X[104X
    [4X[28X      55, 56, 57, 71, 71, 71, 72, 73, 74, 78, 79, 83, 88, 89, 90, 94, [128X[104X
    [4X[28X      100, 101, 105 ] ][128X[104X
    [4X[25Xgap>[125X [27Xfus:= rep[1];;[127X[104X
    [4X[25Xgap>[125X [27XSize( o8p2 ) / (2^9*168);[127X[104X
    [4X[28X2025[128X[104X
    [4X[25Xgap>[125X [27Xsub:= CharacterTable( "2^6:A8" );;[127X[104X
    [4X[25Xgap>[125X [27Xsubfus:= GetFusionMap( sub, o8p2 );[127X[104X
    [4X[28X[ 1, 3, 2, 2, 4, 5, 6, 13, 3, 6, 12, 13, 14, 7, 21, 24, 11, 30, 29, [128X[104X
    [4X[28X  31, 13, 17, 15, 16, 14, 17, 36, 37, 18, 41, 24, 44, 48, 28, 33, 32, [128X[104X
    [4X[28X  34, 35, 35, 51, 51 ][128X[104X
    [4X[25Xgap>[125X [27Xfus:= CompositionMaps( fus, subfus );[127X[104X
    [4X[28X[ 1, 2, 5, 5, 3, 4, 5, 23, 2, 5, 19, 23, 20, 7, 37, 31, 17, 50, 51, [128X[104X
    [4X[28X  43, 23, 23, 21, 22, 20, 23, 56, 57, 24, 72, 31, 78, 89, 52, 45, 44, [128X[104X
    [4X[28X  53, 55, 55, 100, 100 ][128X[104X
    [4X[25Xgap>[125X [27XSize( sub ) / (2^9*168);[127X[104X
    [4X[28X15[128X[104X
    [4X[25Xgap>[125X [27XList( Irr( sub ), Degree );[127X[104X
    [4X[28X[ 1, 7, 14, 20, 21, 21, 21, 28, 35, 45, 45, 56, 64, 70, 28, 28, 35, [128X[104X
    [4X[28X  35, 35, 35, 70, 70, 70, 70, 140, 140, 140, 140, 140, 210, 210, 252, [128X[104X
    [4X[28X  252, 280, 280, 315, 315, 315, 315, 420, 448 ][128X[104X
    [4X[25Xgap>[125X [27Xcand:= PermChars( sub, 15 );[127X[104X
    [4X[28X[ Character( CharacterTable( "2^6:A8" ),[128X[104X
    [4X[28X  [ 15, 15, 15, 7, 7, 7, 7, 7, 3, 3, 3, 3, 3, 0, 0, 0, 3, 3, 3, 3, 3, [128X[104X
    [4X[28X      3, 3, 3, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0 ] ) ][128X[104X
    [4X[25Xgap>[125X [27Xind:= Induced( sub, o8p3, cand, fus );[127X[104X
    [4X[28X[ Character( CharacterTable( "O8+(3)" ),[128X[104X
    [4X[28X  [ 57572775, 59535, 59535, 59535, 3591, 0, 0, 0, 0, 0, 0, 0, 0, 0, [128X[104X
    [4X[28X      0, 0, 2187, 0, 27, 135, 135, 135, 243, 0, 0, 0, 0, 0, 0, 0, 0, [128X[104X
    [4X[28X      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 27, 27, 27, 0, 0, 0, 0, 27, [128X[104X
    [4X[28X      27, 27, 27, 0, 8, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, [128X[104X
    [4X[28X      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, [128X[104X
    [4X[28X      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, [128X[104X
    [4X[28X      0, 0 ] ) ][128X[104X
    [4X[25Xgap>[125X [27Xo8p33:= CharacterTable( "O8+(3).3" );;[127X[104X
    [4X[25Xgap>[125X [27Xinv:= InverseMap( GetFusionMap( o8p3, o8p33 ) );[127X[104X
    [4X[28X[ 1, [ 2, 3, 4 ], 5, 6, [ 7, 8, 9 ], [ 10, 11, 12 ], 13, [128X[104X
    [4X[28X  [ 14, 15, 16 ], 17, 18, 19, [ 20, 21, 22 ], 23, [ 24, 25, 26 ], [128X[104X
    [4X[28X  [ 27, 28, 29 ], 30, [ 31, 32, 33 ], [ 34, 35, 36 ], [ 37, 38, 39 ], [128X[104X
    [4X[28X  [ 40, 41, 42 ], [ 43, 44, 45 ], 46, [ 47, 48, 49 ], 50, [128X[104X
    [4X[28X  [ 51, 52, 53 ], 54, 55, 56, 57, [ 58, 59, 60 ], [ 61, 62, 63 ], 64, [128X[104X
    [4X[28X  [ 65, 66, 67 ], 68, [ 69, 70, 71 ], [ 72, 73, 74 ], [ 75, 76, 77 ], [128X[104X
    [4X[28X  [ 78, 79, 80 ], [ 81, 82, 83 ], 84, 85, [ 86, 87, 88 ], [128X[104X
    [4X[28X  [ 89, 90, 91 ], [ 92, 93, 94 ], 95, 96, [ 97, 98, 99 ], [128X[104X
    [4X[28X  [ 100, 101, 102 ], [ 103, 104, 105 ], [ 106, 107, 108 ], [128X[104X
    [4X[28X  [ 109, 110, 111 ], [ 112, 113, 114 ] ][128X[104X
    [4X[25Xgap>[125X [27Xext:= CompositionMaps( ind[1], inv );[127X[104X
    [4X[28X[ 57572775, 59535, 3591, 0, 0, 0, 0, 0, 2187, 0, 27, 135, 243, 0, 0, [128X[104X
    [4X[28X  0, 0, 0, 0, 0, 27, 0, 0, 27, 27, 0, 8, 1, 1, 0, 0, 0, 0, 0, 0, 0, [128X[104X
    [4X[28X  0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ][128X[104X
    [4X[25Xgap>[125X [27Xperms:= PermChars( o8p33, rec( torso:= ext ) );[127X[104X
    [4X[28X[ Character( CharacterTable( "O8+(3).3" ),[128X[104X
    [4X[28X  [ 57572775, 59535, 3591, 0, 0, 0, 0, 0, 2187, 0, 27, 135, 243, 0, [128X[104X
    [4X[28X      0, 0, 0, 0, 0, 0, 27, 0, 0, 27, 27, 0, 8, 1, 1, 0, 0, 0, 0, 0, [128X[104X
    [4X[28X      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3159, [128X[104X
    [4X[28X      3159, 243, 243, 39, 39, 3, 3, 0, 0, 0, 0, 0, 0, 0, 0, 3, 3, 3, [128X[104X
    [4X[28X      3, 3, 3, 0, 0, 0, 0, 0, 0, 2, 2, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, [128X[104X
    [4X[28X      0, 0 ] ) ][128X[104X
    [4X[25Xgap>[125X [27XPermCharInfo( o8p33, perms ).ATLAS;[127X[104X
    [4X[28X[ "1a+780aabb+2457a+2808abc+9450aaabbcc+18200abcdddef+24192a+54600a^{5\[128X[104X
    [4X[28X}b+70200aabb+87360ab+139776a^{5}+147420a^{4}b^{4}+163800ab+184275aabc+\[128X[104X
    [4X[28X199017aa+218700a+245700a+291200aef+332800a^{4}b^{5}c^{5}+491400aaabcd+\[128X[104X
    [4X[28X531441a^{5}b^{4}c^{4}+552825a^{4}+568620aabb+698880a^{4}b^{4}+716800aa\[128X[104X
    [4X[28Xabbccdddeeff+786240aabb+873600aa+998400aa+1257984a^{6}+1397760aa" ][128X[104X
  [4X[32X[104X
  
  
  [1X8.6 [33X[0;0YThe Action of [22XO_7(3).2[122X[101X[1X on the Cosets of [22X2^7.S_7[122X[101X[1X[133X[101X
  
  [33X[0;0YWe  want to know whether the permutation character of [22XO_7(3).2[122X (see [CCN+85,
  p.  108])  on  the  cosets  of  its  maximal  subgroup  [22XU[122X of type [22X2^7.S_7[122X is
  multiplicity-free.  As in the previous examples, first we try to compute the
  permutation  character  of  the  simple  group [22XO_7(3)[122X. It turns out that the
  direct computation of all candidates from the degree is very time consuming.
  But  we  can use for example the additional information provided by the fact
  that [22XU[122X contains an [22XA_7[122X subgroup. We compute the possible class fusions.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xo73:= CharacterTable( "O7(3)" );;[127X[104X
    [4X[25Xgap>[125X [27Xa7:= CharacterTable( "A7" );;[127X[104X
    [4X[25Xgap>[125X [27Xfus:= PossibleClassFusions( a7, o73 );[127X[104X
    [4X[28X[ [ 1, 3, 6, 10, 15, 16, 24, 33, 33 ], [128X[104X
    [4X[28X  [ 1, 3, 7, 10, 15, 16, 22, 33, 33 ] ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YWe  cannot decide easily which fusion is the right one, but already the fact
  that  no  other  fusions  are  possible  gives  us  some  information  about
  impossible constituents of the permutation character we want to compute.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xind:= List( fus,[127X[104X
    [4X[25X>[125X [27X      x -> Induced( a7, o73, [ TrivialCharacter( a7 ) ], x )[1] );;[127X[104X
    [4X[25Xgap>[125X [27Xmat:= MatScalarProducts( o73, Irr( o73 ), ind );;[127X[104X
    [4X[25Xgap>[125X [27Xsum:= Sum( mat );[127X[104X
    [4X[28X[ 2, 6, 2, 0, 8, 6, 2, 4, 4, 8, 3, 0, 4, 4, 9, 3, 5, 0, 0, 9, 0, 10, [128X[104X
    [4X[28X  5, 6, 15, 1, 12, 1, 15, 7, 2, 4, 14, 16, 0, 12, 12, 7, 8, 8, 14, [128X[104X
    [4X[28X  12, 12, 14, 6, 6, 20, 16, 12, 12, 12, 10, 10, 12, 12, 8, 12, 6 ][128X[104X
    [4X[25Xgap>[125X [27Xconst:= Filtered( [ 1 .. Length( sum ) ], x -> sum[x] <> 0 );[127X[104X
    [4X[28X[ 1, 2, 3, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 20, 22, 23, 24, [128X[104X
    [4X[28X  25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 36, 37, 38, 39, 40, 41, 42, [128X[104X
    [4X[28X  43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58 ][128X[104X
    [4X[25Xgap>[125X [27XLength( const );[127X[104X
    [4X[28X52[128X[104X
    [4X[25Xgap>[125X [27Xconst:= Irr( o73 ){ const };;[127X[104X
    [4X[25Xgap>[125X [27Xrat:= RationalizedMat( const );;[127X[104X
  [4X[32X[104X
  
  [33X[0;0YBut  much  more  can  be  deduced  from  the  fact that certain zeros of the
  permutation character can be predicted.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xnames:= ClassNames( o73 );[127X[104X
    [4X[28X[ "1a", "2a", "2b", "2c", "3a", "3b", "3c", "3d", "3e", "3f", "3g", [128X[104X
    [4X[28X  "4a", "4b", "4c", "4d", "5a", "6a", "6b", "6c", "6d", "6e", "6f", [128X[104X
    [4X[28X  "6g", "6h", "6i", "6j", "6k", "6l", "6m", "6n", "6o", "6p", "7a", [128X[104X
    [4X[28X  "8a", "8b", "9a", "9b", "9c", "9d", "10a", "10b", "12a", "12b", [128X[104X
    [4X[28X  "12c", "12d", "12e", "12f", "12g", "12h", "13a", "13b", "14a", [128X[104X
    [4X[28X  "15a", "18a", "18b", "18c", "18d", "20a" ][128X[104X
    [4X[25Xgap>[125X [27XList( fus, x -> names{ x } );[127X[104X
    [4X[28X[ [ "1a", "2b", "3b", "3f", "4d", "5a", "6h", "7a", "7a" ], [128X[104X
    [4X[28X  [ "1a", "2b", "3c", "3f", "4d", "5a", "6f", "7a", "7a" ] ][128X[104X
    [4X[25Xgap>[125X [27Xtorso:= [ 28431 ];;[127X[104X
    [4X[25Xgap>[125X [27Xzeros:= [ 5, 8, 9, 11, 17, 20, 23, 28, 29, 32, 36, 37, 38,[127X[104X
    [4X[25X>[125X [27X             43, 46, 47, 48, 53, 54, 55, 56, 57, 58 ];;[127X[104X
    [4X[25Xgap>[125X [27Xnames{ zeros };[127X[104X
    [4X[28X[ "3a", "3d", "3e", "3g", "6a", "6d", "6g", "6l", "6m", "6p", "9a", [128X[104X
    [4X[28X  "9b", "9c", "12b", "12e", "12f", "12g", "15a", "18a", "18b", "18c", [128X[104X
    [4X[28X  "18d", "20a" ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YEvery  order  [22X3[122X  element  of  [22XU[122X  lies  in an [22XA_7[122X subgroup of [22XU[122X, so among the
  classes  of  element  order  [22X3[122X,  at most the classes [10X3B[110X, [10X3C[110X, and [10X3F[110X can have
  nonzero  permutation character values. The excluded classes of element order
  [22X6[122X  are the square roots of the excluded order [22X3[122X elements, likewise the given
  classes of element orders [22X9[122X, [22X12[122X, and [22X18[122X are excluded. The character value on
  [10X20A[110X  must  be  zero because [22XU[122X does not contain elements of this order. So we
  enter the additional information about these zeros.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xfor i in zeros do[127X[104X
    [4X[25X>[125X [27X     torso[i]:= 0;[127X[104X
    [4X[25X>[125X [27X   od;[127X[104X
    [4X[25Xgap>[125X [27Xtorso;[127X[104X
    [4X[28X[ 28431,,,, 0,,, 0, 0,, 0,,,,,, 0,,, 0,,, 0,,,,, 0, 0,,, 0,,,, 0, 0, [128X[104X
    [4X[28X  0,,,,, 0,,, 0, 0, 0,,,,, 0, 0, 0, 0, 0, 0 ][128X[104X
    [4X[25Xgap>[125X [27Xperms:= PermChars( o73, rec( torso:= torso, chars:= rat ) );[127X[104X
    [4X[28X[ Character( CharacterTable( "O7(3)" ),[128X[104X
    [4X[28X  [ 28431, 567, 567, 111, 0, 0, 243, 0, 0, 81, 0, 15, 3, 27, 15, 6, [128X[104X
    [4X[28X      0, 0, 27, 0, 3, 27, 0, 0, 0, 3, 9, 0, 0, 3, 3, 0, 4, 1, 1, 0, [128X[104X
    [4X[28X      0, 0, 0, 2, 2, 3, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, [128X[104X
    [4X[28X      0 ] ) ][128X[104X
    [4X[25Xgap>[125X [27XPermCharInfo( o73, perms ).ATLAS;[127X[104X
    [4X[28X[ "1a+78a+168a+182a+260ab+1092a+2457a+2730a+4095b+5460a+11648a" ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YWe  see that this character is already multiplicity free, so this holds also
  for  its  extension to [22XO_7(3).2[122X, and we need not compute this extension. (Of
  course we could compute it in the same way as in the examples above.)[133X
  
  
  [1X8.7 [33X[0;0YThe Action of [22XO_8^+(3).2_1[122X[101X[1X on the Cosets of [22X2^7.A_8[122X[101X[1X[133X[101X
  
  [33X[0;0YWe  are  interested  in  the  permutation  character  of  [22XO_8^+(3).2_1[122X  that
  corresponds  to  the action on the cosets of a subgroup of type [22X2^7.A_8[122X. The
  intersection  of  the  point stabilizer with the simple group [22XO_8^+(3)[122X is of
  type  [22X2^6.A_8[122X.  First  we  compute  the class fusion of these groups, modulo
  problems with ambiguities due to table automorphisms.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xo8p3:= CharacterTable( "O8+(3)" );;[127X[104X
    [4X[25Xgap>[125X [27Xo8p2:= CharacterTable( "O8+(2)" );;[127X[104X
    [4X[25Xgap>[125X [27Xfus:= PossibleClassFusions( o8p2, o8p3 );;[127X[104X
    [4X[25Xgap>[125X [27XNamesOfFusionSources( o8p2 );[127X[104X
    [4X[28X[ "A9", "2^8:O8+(2)", "(D10xD10).2^2", "(3x3^3:S3):2", [128X[104X
    [4X[28X  "(3x3^(1+2)+:2A4).2", "2^(3+3+3).L3(2)", "NRS(O8+(2),2^(3+3+3)_a)", [128X[104X
    [4X[28X  "NRS(O8+(2),2^(3+3+3)_b)", "O8+(2)N2", "O8+(2)M2", "O8+(2)M3", [128X[104X
    [4X[28X  "O8+(2)M5", "O8+(2)M6", "O8+(2)M8", "O8+(2)M9", "(3xU4(2)):2", [128X[104X
    [4X[28X  "O8+(2)M11", "O8+(2)M12", "2^(1+8)_+:(S3xS3xS3)", "3^4:2^3.S4(a)", [128X[104X
    [4X[28X  "(A5xA5):2^2", "O8+(2)M16", "O8+(2)M17", "2^(1+8)+.O8+(2)", "7:6", [128X[104X
    [4X[28X  "(A5xD10).2", "(D10xA5).2", "O8+(2)N5C", "2^6:A8", "2.O8+(2)", [128X[104X
    [4X[28X  "2^2.O8+(2)", "S6(2)" ][128X[104X
    [4X[25Xgap>[125X [27Xsub:= CharacterTable( "2^6:A8" );;[127X[104X
    [4X[25Xgap>[125X [27Xsubfus:= GetFusionMap( sub, o8p2 );[127X[104X
    [4X[28X[ 1, 3, 2, 2, 4, 5, 6, 13, 3, 6, 12, 13, 14, 7, 21, 24, 11, 30, 29, [128X[104X
    [4X[28X  31, 13, 17, 15, 16, 14, 17, 36, 37, 18, 41, 24, 44, 48, 28, 33, 32, [128X[104X
    [4X[28X  34, 35, 35, 51, 51 ][128X[104X
    [4X[25Xgap>[125X [27Xfus:= List( fus, x -> CompositionMaps( x, subfus ) );;[127X[104X
    [4X[25Xgap>[125X [27Xfus:= Set( fus );;[127X[104X
    [4X[25Xgap>[125X [27XLength( fus );[127X[104X
    [4X[28X24[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  ambiguities  due  to Galois automorphisms disappear when we are looking
  for the permutation characters induced by the fusions.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xind:= List( fus, x -> Induced( sub, o8p3,[127X[104X
    [4X[25X>[125X [27X                             [ TrivialCharacter( sub ) ], x )[1] );;[127X[104X
    [4X[25Xgap>[125X [27Xind:= Set( ind );;[127X[104X
    [4X[25Xgap>[125X [27XLength( ind );[127X[104X
    [4X[28X6[128X[104X
  [4X[32X[104X
  
  [33X[0;0YNow  we  try  to  extend  the  candidates to [22XO_8^+(3).2_1[122X; the choice of the
  fusion  of [22XO_8^+(3)[122X into [22XO_8^+(3).2_1[122X determines which of the candidates may
  extend.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xo8p32:= CharacterTable( "O8+(3).2_1" );;[127X[104X
    [4X[25Xgap>[125X [27Xfus:= GetFusionMap( o8p3, o8p32 );;[127X[104X
    [4X[25Xgap>[125X [27Xext:= List( ind, x -> CompositionMaps( x, InverseMap( fus ) ) );;[127X[104X
    [4X[25Xgap>[125X [27Xext:= Filtered( ext, x -> ForAll( x, IsInt ) );[127X[104X
    [4X[28X[ [ 3838185, 17577, 8505, 8505, 873, 0, 0, 0, 0, 6561, 0, 0, 729, 0, [128X[104X
    [4X[28X      9, 105, 45, 45, 105, 30, 0, 0, 0, 0, 0, 0, 0, 0, 0, 189, 0, 0, [128X[104X
    [4X[28X      0, 9, 9, 27, 27, 0, 0, 27, 9, 0, 8, 1, 1, 0, 0, 0, 0, 0, 0, 0, [128X[104X
    [4X[28X      0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 9, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, [128X[104X
    [4X[28X      0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0 ], [128X[104X
    [4X[28X  [ 3838185, 17577, 8505, 8505, 873, 0, 6561, 0, 0, 0, 0, 0, 729, 0, [128X[104X
    [4X[28X      9, 105, 45, 45, 105, 30, 0, 0, 0, 0, 0, 0, 189, 0, 0, 0, 9, 0, [128X[104X
    [4X[28X      0, 0, 9, 27, 27, 0, 0, 9, 27, 0, 8, 1, 1, 0, 0, 0, 0, 0, 0, 0, [128X[104X
    [4X[28X      0, 2, 0, 0, 0, 0, 0, 9, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, [128X[104X
    [4X[28X      0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ] ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YWe  compute  the  extensions  of  the  first candidate; the other belongs to
  another class of subgroups, which is the image under an outer automorphism.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xperms:= PermChars( o8p32, rec( torso:= ext[1] ) );[127X[104X
    [4X[28X[ Character( CharacterTable( "O8+(3).2_1" ),[128X[104X
    [4X[28X  [ 3838185, 17577, 8505, 8505, 873, 0, 0, 0, 0, 6561, 0, 0, 729, 0, [128X[104X
    [4X[28X      9, 105, 45, 45, 105, 30, 0, 0, 0, 0, 0, 0, 0, 0, 0, 189, 0, 0, [128X[104X
    [4X[28X      0, 9, 9, 27, 27, 0, 0, 27, 9, 0, 8, 1, 1, 0, 0, 0, 0, 0, 0, 0, [128X[104X
    [4X[28X      0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 9, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, [128X[104X
    [4X[28X      0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 3159, 1575, 567, 63, 87, [128X[104X
    [4X[28X      15, 0, 0, 45, 0, 81, 9, 27, 0, 0, 3, 3, 3, 3, 5, 5, 0, 0, 0, 4, [128X[104X
    [4X[28X      0, 0, 27, 0, 9, 0, 0, 15, 0, 3, 0, 0, 2, 0, 0, 0, 0, 0, 3, 0, [128X[104X
    [4X[28X      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ] ) ][128X[104X
    [4X[25Xgap>[125X [27XPermCharInfo( o8p32, perms ).ATLAS;[127X[104X
    [4X[28X[ "1a+260abc+520ab+819a+2808b+9450aab+18200a+23400ac+29120b+36400aab+4\[128X[104X
    [4X[28X6592abce+49140d+66339a+98280ab+163800a+189540d+232960d+332800ab+368550\[128X[104X
    [4X[28Xa+419328a+531441ab" ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YNow we repeat the calculations for [22XO_8^+(3).2_2[122X instead of [22XO_8^+(3).2_1[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xo8p32:= CharacterTable( "O8+(3).2_2" );;[127X[104X
    [4X[25Xgap>[125X [27Xfus:= GetFusionMap( o8p3, o8p32 );;[127X[104X
    [4X[25Xgap>[125X [27Xext:= List( ind, x -> CompositionMaps( x, InverseMap( fus ) ) );;[127X[104X
    [4X[25Xgap>[125X [27Xext:= Filtered( ext, x -> ForAll( x, IsInt ) );;[127X[104X
    [4X[25Xgap>[125X [27Xperms:= PermChars( o8p32, rec( torso:= ext[1] ) );[127X[104X
    [4X[28X[ Character( CharacterTable( "O8+(3).2_2" ),[128X[104X
    [4X[28X  [ 3838185, 17577, 8505, 873, 0, 0, 0, 6561, 0, 0, 0, 0, 729, 0, 9, [128X[104X
    [4X[28X      105, 45, 105, 30, 0, 0, 0, 0, 0, 0, 189, 0, 0, 0, 9, 0, 9, 27, [128X[104X
    [4X[28X      0, 0, 0, 27, 27, 9, 0, 8, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, [128X[104X
    [4X[28X      2, 0, 0, 0, 0, 0, 9, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, [128X[104X
    [4X[28X      0, 6, 0, 0, 0, 0, 0, 0, 0, 199017, 2025, 297, 441, 73, 9, 0, [128X[104X
    [4X[28X      1215, 0, 0, 0, 0, 0, 81, 0, 0, 0, 0, 27, 27, 0, 1, 9, 12, 0, 0, [128X[104X
    [4X[28X      45, 0, 0, 1, 0, 0, 3, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 1, 0, [128X[104X
    [4X[28X      0, 0, 0, 0, 0 ] ) ][128X[104X
    [4X[25Xgap>[125X [27XPermCharInfo( o8p32, perms ).ATLAS;[127X[104X
    [4X[28X[ "1a+260aac+520ab+819a+2808a+9450aaa+18200accee+23400ac+29120a+36400a\[128X[104X
    [4X[28X+46592aa+49140c+66339a+93184a+98280ab+163800a+184275ac+189540c+232960c\[128X[104X
    [4X[28X+332800aa+419328a+531441aa" ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YWe  might  be interested in the extension to [22XO_8^+(3).(2^2)_122[122X. It is clear
  that this cannot be multiplicity free because of the multiplicity [10X9450aaa[110X in
  the  character induced from [22XO_8^+(3).2_2[122X. We could put the extensions to the
  index  two  subgroups together, but it is simpler (and not expensive) to run
  the same program as above.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xo8p322:= CharacterTable( "O8+(3).(2^2)_{122}" );;[127X[104X
    [4X[25Xgap>[125X [27Xfus:= GetFusionMap( o8p32, o8p322 );;[127X[104X
    [4X[25Xgap>[125X [27Xext:= List( perms, x -> CompositionMaps( x, InverseMap( fus ) ) );;[127X[104X
    [4X[25Xgap>[125X [27Xext:= Filtered( ext, x -> ForAll( x, IsInt ) );;[127X[104X
    [4X[25Xgap>[125X [27Xperms:= PermChars( o8p322, rec( torso:= ext[1] ) );[127X[104X
    [4X[28X[ Character( CharacterTable( "O8+(3).(2^2)_{122}" ),[128X[104X
    [4X[28X  [ 3838185, 17577, 8505, 873, 0, 0, 0, 6561, 0, 0, 729, 0, 9, 105, [128X[104X
    [4X[28X      45, 105, 30, 0, 0, 0, 0, 0, 0, 189, 0, 0, 9, 9, 27, 0, 0, 27, [128X[104X
    [4X[28X      9, 0, 8, 1, 1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 9, 0, 0, [128X[104X
    [4X[28X      0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 3159, 1575, [128X[104X
    [4X[28X      567, 63, 87, 15, 0, 0, 45, 0, 81, 9, 27, 0, 0, 3, 3, 3, 5, 0, [128X[104X
    [4X[28X      0, 4, 0, 0, 27, 0, 9, 0, 0, 15, 0, 3, 0, 0, 2, 0, 0, 0, 3, 0, [128X[104X
    [4X[28X      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 199017, 2025, 297, 441, 73, 9, 0, [128X[104X
    [4X[28X      1215, 0, 0, 0, 0, 81, 0, 0, 0, 27, 27, 0, 1, 9, 12, 0, 0, 45, [128X[104X
    [4X[28X      0, 0, 1, 0, 0, 3, 1, 0, 0, 0, 0, 0, 0, 0, 2, 1, 0, 0, 0, 0, 0, [128X[104X
    [4X[28X      0, 28431, 1647, 135, 63, 87, 39, 0, 0, 243, 27, 0, 0, 81, 63, [128X[104X
    [4X[28X      0, 0, 0, 9, 0, 3, 3, 6, 2, 0, 0, 0, 9, 0, 0, 3, 3, 3, 0, 4, 0, [128X[104X
    [4X[28X      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0 ] ) ][128X[104X
    [4X[25Xgap>[125X [27XPermCharInfo( o8p322, perms ).ATLAS;[127X[104X
    [4X[28X[ "1a+260ace+819a+1040a+2808c+9450aac+18200a+23400ae+29120c+36400aac+4\[128X[104X
    [4X[28X6592ac+49140g+66339a+93184a+163800b+189540g+196560a+232960g+332800ac+3\[128X[104X
    [4X[28X68550a+419328a+531441ac" ][128X[104X
  [4X[32X[104X
  
  
  [1X8.8 [33X[0;0YThe Action of [22XS_4(4).4[122X[101X[1X on the Cosets of [22X5^2.[2^5][122X[101X[1X[133X[101X
  
  [33X[0;0YWe  want  to  know  whether  the  permutation character corresponding to the
  action  of  [22XS_4(4).4[122X  (see [CCN+85,  p.  44])  on  the cosets of its maximal
  subgroup  of  type  [22X5^2:[2^5][122X  is  multiplicity  free.  The library names of
  subgroups  for which the class fusions are stored are listed as value of the
  attribute  [2XNamesOfFusionSources[102X  ([14XReference:  NamesOfFusionSources[114X), and for
  groups  whose  isomorphism  type  is  not determined by the name this is the
  recommended  way  to find out whether the table of the subgroup is contained
  in  the  [5XGAP[105X  library and known to belong to this group. (It might be that a
  table  with  such  a name is contained in the library but belongs to another
  group,  and  it  may also be that the table of the group is contained in the
  library  --with any name-- but it is not known that this group is isomorphic
  to a subgroup of [22XS_4(4).4[122X.)[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xs444:= CharacterTable( "S4(4).4" );;[127X[104X
    [4X[25Xgap>[125X [27XNamesOfFusionSources( s444 );[127X[104X
    [4X[28X[ "(L3(2)xS4(4):2).2", "S4(4)", "S4(4).2" ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YSo  we  cannot  simply  fetch  the table of the subgroup. As in the previous
  examples, we compute the possible permutation characters.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xperms:= PermChars( s444,[127X[104X
    [4X[25X>[125X [27X               rec( torso:= [ Size( s444 ) / ( 5^2*2^5 ) ] ) );[127X[104X
    [4X[28X[ Character( CharacterTable( "S4(4).4" ),[128X[104X
    [4X[28X  [ 4896, 384, 96, 0, 16, 32, 36, 16, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, [128X[104X
    [4X[28X      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ] ), [128X[104X
    [4X[28X  Character( CharacterTable( "S4(4).4" ),[128X[104X
    [4X[28X  [ 4896, 192, 32, 0, 0, 8, 6, 1, 0, 2, 0, 0, 36, 0, 12, 0, 0, 0, 1, [128X[104X
    [4X[28X      0, 6, 6, 2, 2, 0, 0, 0, 0, 1, 1 ] ), [128X[104X
    [4X[28X  Character( CharacterTable( "S4(4).4" ),[128X[104X
    [4X[28X  [ 4896, 240, 64, 0, 8, 8, 36, 16, 0, 0, 0, 0, 0, 12, 8, 0, 4, 4, 0, [128X[104X
    [4X[28X      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ] ) ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YSo there are three candidates. None of them is multiplicity free, so we need
  not  decide  which of the candidates actually belongs to the group [22X5^2:[2^5][122X
  we have in mind.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XPermCharInfo( s444, perms ).ATLAS;[127X[104X
    [4X[28X[ "1abcd+50abcd+153abcd+170a^{4}b^{4}+680aabb", [128X[104X
    [4X[28X  "1a+50ac+153a+170aab+256a+680abb+816a+1020a", [128X[104X
    [4X[28X  "1ac+50ac+68a+153abcd+170aabbb+204a+680abb+1020a" ][128X[104X
  [4X[32X[104X
  
  [33X[0;0Y(If  we  would  be interested which candidate is the right one, we could for
  example  look  at the intersection with [22XS_4(4)[122X, and hope for a contradiction
  to the fact that the group must lie in a [22X(A_5 × A_5):2[122X subgroup.)[133X
  
  
  [1X8.9 [33X[0;0YThe Action of [22XCo_1[122X[101X[1X on the Cosets of Involution Centralizers[133X[101X
  
  [33X[0;0YWe  compute  the  permutation characters of the sporadic simple Conway group
  [22XCo_1[122X  (see [CCN+85,  p.  180]) corresponding to the actions on the cosets of
  involution  centralizers.  Equivalently,  we are interested in the action of
  [22XCo_1[122X  on  conjugacy classes of involutions. These characters can be computed
  as follows. First we take the table of [22XCo_1[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xt:= CharacterTable( "Co1" );[127X[104X
    [4X[28XCharacterTable( "Co1" )[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe centralizer of each [10X2A[110X element is a maximal subgroup of [22XCo_1[122X. This group
  is  also  contained  in the table library. So we can compute the permutation
  character  by  explicit  induction, and the decomposition in irreducibles is
  computed with the command [2XPermCharInfo[102X ([14XReference: PermCharInfo[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xs:= CharacterTable( Maxes( t )[5] );[127X[104X
    [4X[28XCharacterTable( "2^(1+8)+.O8+(2)" )[128X[104X
    [4X[25Xgap>[125X [27Xind:= Induced( s, t, [ TrivialCharacter( s ) ] );;[127X[104X
    [4X[25Xgap>[125X [27XPermCharInfo( t, ind ).ATLAS;[127X[104X
    [4X[28X[ "1a+299a+17250a+27300a+80730a+313950a+644644a+2816856a+5494125a+1243\[128X[104X
    [4X[28X2420a+24794000a" ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  centralizer  of  a  [10X2B[110X  element  is not maximal. First we compute which
  maximal  subgroup  can  contain  it.  The  character  tables  of all maximal
  subgroups  of  [22XCo_1[122X are contained in the [5XGAP[105X's table library, so we may take
  these tables and look at the group orders.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xcentorder:= SizesCentralizers( t )[3];;[127X[104X
    [4X[25Xgap>[125X [27Xmaxes:= List( Maxes( t ), CharacterTable );;[127X[104X
    [4X[25Xgap>[125X [27Xcand:= Filtered( maxes, x -> Size( x ) mod centorder = 0 );[127X[104X
    [4X[28X[ CharacterTable( "(A4xG2(4)):2" ) ][128X[104X
    [4X[25Xgap>[125X [27Xu:= cand[1];;[127X[104X
    [4X[25Xgap>[125X [27Xindex:= Size( u ) / centorder;[127X[104X
    [4X[28X3[128X[104X
  [4X[32X[104X
  
  [33X[0;0YSo  there  is a unique class of maximal subgroups containing the centralizer
  of a [10X2B[110X element, as a subgroup of index [22X3[122X. We compute the unique permutation
  character of degree [22X3[122X of this group, and induce this character to [22XG[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xsubperm:= PermChars( u, rec( degree := index, bounds := false ) );[127X[104X
    [4X[28X[ Character( CharacterTable( "(A4xG2(4)):2" ),[128X[104X
    [4X[28X  [ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, [128X[104X
    [4X[28X      3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 3, 3, [128X[104X
    [4X[28X      3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, [128X[104X
    [4X[28X      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, [128X[104X
    [4X[28X      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, [128X[104X
    [4X[28X      1, 1, 1, 1, 1, 1 ] ) ][128X[104X
    [4X[25Xgap>[125X [27Xsubperm = PermChars( u, rec( torso := [ 3 ] ) );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xind:= Induced( u, t, subperm );[127X[104X
    [4X[28X[ Character( CharacterTable( "Co1" ),[128X[104X
    [4X[28X  [ 2065694400, 181440, 119408, 38016, 2779920, 0, 0, 378, 30240, [128X[104X
    [4X[28X      864, 0, 720, 316, 80, 2520, 30, 0, 6480, 1508, 0, 0, 0, 0, 0, [128X[104X
    [4X[28X      38, 18, 105, 0, 600, 120, 56, 24, 0, 12, 0, 0, 0, 120, 48, 18, [128X[104X
    [4X[28X      0, 0, 6, 0, 360, 144, 108, 0, 0, 10, 0, 0, 0, 0, 0, 4, 2, 3, 9, [128X[104X
    [4X[28X      0, 0, 15, 3, 0, 0, 4, 4, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, [128X[104X
    [4X[28X      12, 8, 0, 6, 0, 0, 3, 0, 1, 0, 3, 3, 0, 0, 0, 0, 0, 0, 0, 0, 3, [128X[104X
    [4X[28X      0 ] ) ][128X[104X
    [4X[25Xgap>[125X [27XPermCharInfo( t, ind ).ATLAS;[127X[104X
    [4X[28X[ "1a+1771a+8855a+27300aa+313950a+345345a+644644aa+871884aaa+1771000a+\[128X[104X
    [4X[28X2055625a+4100096a+7628985a+9669660a+12432420aa+21528000aa+23244375a+24\[128X[104X
    [4X[28X174150aa+24794000a+31574400aa+40370176a+60435375a+85250880aa+100725625\[128X[104X
    [4X[28Xa+106142400a+150732800a+184184000a+185912496a+207491625a+299710125a+30\[128X[104X
    [4X[28X2176875a" ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YFinally, we try the same for the centralizer of a [10X2C[110X element.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xcentorder:= SizesCentralizers( t )[4];;[127X[104X
    [4X[25Xgap>[125X [27Xcand:= Filtered( maxes, x -> Size( x ) mod centorder = 0 );[127X[104X
    [4X[28X[ CharacterTable( "Co2" ), CharacterTable( "2^11:M24" ) ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  group  order  excludes all except two classes of maximal subgroups. But
  the [10X2C[110X centralizer cannot lie in [22XCo_2[122X because the involution centralizers in
  [22XCo_2[122X are too small.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xu:= cand[1];;[127X[104X
    [4X[25Xgap>[125X [27XGetFusionMap( u, t );[127X[104X
    [4X[28X[ 1, 2, 2, 4, 7, 6, 9, 11, 11, 10, 11, 12, 14, 17, 16, 21, 23, 20, [128X[104X
    [4X[28X  22, 22, 24, 28, 30, 33, 31, 32, 33, 33, 37, 42, 41, 43, 44, 48, 52, [128X[104X
    [4X[28X  49, 53, 55, 53, 52, 54, 60, 60, 60, 64, 65, 65, 67, 66, 70, 73, 72, [128X[104X
    [4X[28X  78, 79, 84, 85, 87, 92, 93, 93 ][128X[104X
    [4X[25Xgap>[125X [27Xcentorder;[127X[104X
    [4X[28X389283840[128X[104X
    [4X[25Xgap>[125X [27XSizesCentralizers( u )[4];[127X[104X
    [4X[28X1474560[128X[104X
  [4X[32X[104X
  
  [33X[0;0YSo we try the second candidate.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xu:= cand[2];[127X[104X
    [4X[28XCharacterTable( "2^11:M24" )[128X[104X
    [4X[25Xgap>[125X [27Xindex:= Size( u ) / centorder;[127X[104X
    [4X[28X1288[128X[104X
    [4X[25Xgap>[125X [27Xsubperm:= PermChars( u, rec( torso := [ index ] ) );[127X[104X
    [4X[28X[ Character( CharacterTable( "2^11:M24" ),[128X[104X
    [4X[28X  [ 1288, 1288, 1288, 56, 56, 56, 56, 56, 56, 48, 48, 48, 48, 48, 10, [128X[104X
    [4X[28X      10, 10, 10, 7, 7, 8, 8, 8, 8, 8, 8, 4, 4, 4, 4, 4, 4, 4, 4, 4, [128X[104X
    [4X[28X      4, 4, 3, 3, 3, 2, 2, 2, 2, 2, 2, 3, 3, 3, 0, 0, 0, 0, 2, 2, 2, [128X[104X
    [4X[28X      2, 3, 3, 3, 1, 1, 2, 2, 2, 2, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, [128X[104X
    [4X[28X      0, 0, 0 ] ) ][128X[104X
    [4X[25Xgap>[125X [27Xsubperm = PermChars( u, rec( degree:= index, bounds := false ) );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xind:= Induced( u, t, subperm );[127X[104X
    [4X[28X[ Character( CharacterTable( "Co1" ),[128X[104X
    [4X[28X  [ 10680579000, 1988280, 196560, 94744, 0, 17010, 0, 945, 7560, [128X[104X
    [4X[28X      3432, 2280, 1728, 252, 308, 0, 225, 0, 0, 0, 270, 0, 306, 0, [128X[104X
    [4X[28X      46, 45, 25, 0, 0, 120, 32, 12, 52, 36, 36, 0, 0, 0, 0, 0, 45, [128X[104X
    [4X[28X      15, 0, 9, 3, 0, 0, 0, 0, 18, 0, 30, 0, 6, 18, 0, 3, 5, 0, 0, 0, [128X[104X
    [4X[28X      0, 0, 0, 0, 0, 2, 2, 0, 0, 0, 0, 3, 0, 0, 0, 0, 1, 0, 0, 0, 0, [128X[104X
    [4X[28X      6, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ] ) ][128X[104X
    [4X[25Xgap>[125X [27XPermCharInfo( t, ind ).ATLAS;[127X[104X
    [4X[28X[ "1a+17250aa+27300a+80730aa+644644aaa+871884a+1821600a+2055625aaa+281\[128X[104X
    [4X[28X6856a+5494125a^{4}+12432420aa+16347825aa+23244375a+24174150aa+24667500\[128X[104X
    [4X[28Xaa+24794000aaa+31574400a+40370176a+55255200a+66602250a^{4}+83720000aa+\[128X[104X
    [4X[28X85250880aaa+91547820aa+106142400a+150732800a+184184000aaa+185912496aaa\[128X[104X
    [4X[28X+185955000aaa+207491625aaa+215547904aa+241741500aaa+247235625a+2578576\[128X[104X
    [4X[28X00aa+259008750a+280280000a+302176875a+326956500a+387317700a+402902500a\[128X[104X
    [4X[28X+464257024a+469945476b+502078500a+503513010a+504627200a+522161640a" ][128X[104X
  [4X[32X[104X
  
  
  [1X8.10 [33X[0;0YThe Multiplicity Free Permutation Characters of [22XG_2(3)[122X[101X[1X[133X[101X
  
  [33X[0;0YWe  compute  the multiplicity free possible permutation characters of [22XG_2(3)[122X
  (see [CCN+85, p. 60]). For each divisor [22Xd[122X of the group order, we compute all
  those  possible  permutation  characters  of  degree  [22Xd[122X  of [22XG[122X for which each
  irreducible  constituent occurs with multiplicity at most [22X1[122X; this is done by
  prescribing  the  [10Xmaxmult[110X  component  of  the  second  argument of [2XPermChars[102X
  ([14XReference: PermChars[114X) to be the list with [22X1[122X at each position.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xt:= CharacterTable( "G2(3)" );[127X[104X
    [4X[28XCharacterTable( "G2(3)" )[128X[104X
    [4X[25Xgap>[125X [27Xt:= CharacterTable( "G2(3)" );;[127X[104X
    [4X[25Xgap>[125X [27Xn:= Length( RationalizedMat( Irr( t ) ) );;[127X[104X
    [4X[25Xgap>[125X [27Xmaxmult:= List( [ 1 .. n ], i -> 1 );;[127X[104X
    [4X[25Xgap>[125X [27Xperms:= [];;[127X[104X
    [4X[25Xgap>[125X [27Xdivs:= DivisorsInt( Size( t ) );;[127X[104X
    [4X[25Xgap>[125X [27Xfor d in divs do[127X[104X
    [4X[25X>[125X [27X     Append( perms,[127X[104X
    [4X[25X>[125X [27X             PermChars( t, rec( bounds  := false,[127X[104X
    [4X[25X>[125X [27X                                degree  := d,[127X[104X
    [4X[25X>[125X [27X                                maxmult := maxmult ) ) );[127X[104X
    [4X[25X>[125X [27X   od;[127X[104X
    [4X[25Xgap>[125X [27XLength( perms );[127X[104X
    [4X[28X42[128X[104X
    [4X[25Xgap>[125X [27XList( perms, Degree );[127X[104X
    [4X[28X[ 1, 351, 351, 364, 364, 378, 378, 546, 546, 546, 546, 546, 702, 702, [128X[104X
    [4X[28X  728, 728, 1092, 1092, 1092, 1092, 1092, 1092, 1092, 1092, 1456, [128X[104X
    [4X[28X  1456, 1638, 1638, 2184, 2184, 2457, 2457, 2457, 2457, 3159, 3276, [128X[104X
    [4X[28X  3276, 3276, 3276, 4368, 6552, 6552 ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YFor finding out which of these candidates are really permutation characters,
  we could inspect them piece by piece, using the information in [CCN+85]. For
  example,  the  candidates  of degrees [22X351[122X, [22X364[122X, and [22X378[122X are induced from the
  trivial  characters  of  maximal  subgroups  of [22XG[122X, whereas the candidates of
  degree [22X546[122X are not permutation characters.[133X
  
  [33X[0;0YSince  the  table  of  marks  of  [22XG[122X  is available in [5XGAP[105X, we can extract all
  permutation  characters  from  the  table  of marks, and then filter out the
  multiplicity free ones.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xtom:= TableOfMarks( "G2(3)" );[127X[104X
    [4X[28XTableOfMarks( "G2(3)" )[128X[104X
    [4X[25Xgap>[125X [27Xtbl:= CharacterTable( "G2(3)" );[127X[104X
    [4X[28XCharacterTable( "G2(3)" )[128X[104X
    [4X[25Xgap>[125X [27Xpermstom:= PermCharsTom( tbl, tom );;[127X[104X
    [4X[25Xgap>[125X [27XLength( permstom );[127X[104X
    [4X[28X433[128X[104X
    [4X[25Xgap>[125X [27Xmultfree:= Intersection( perms, permstom );;[127X[104X
    [4X[25Xgap>[125X [27XLength( multfree );[127X[104X
    [4X[28X15[128X[104X
    [4X[25Xgap>[125X [27XList( multfree, Degree );[127X[104X
    [4X[28X[ 1, 351, 351, 364, 364, 378, 378, 702, 702, 728, 728, 1092, 1092, [128X[104X
    [4X[28X  2184, 2184 ][128X[104X
  [4X[32X[104X
  
  
  [1X8.11 [33X[0;0YDegree [22X11200[122X[101X[1X Permutation Characters of [22XO_8^+(2)[122X[101X[1X[133X[101X
  
  [33X[0;0YWe  compute the primitive permutation characters of degree [22X11200[122X of [22XO_8^+(2)[122X
  and  [22XO_8^+(2).2[122X  (see [CCN+85,  p.  85]). The character table of the maximal
  subgroup  of  type [22X3^4:2^3.S_4[122X in [22XO_8^+(2)[122X is not available in the [5XGAP[105X table
  library.  But  the  group  extends to a wreath product of [22XS_3[122X and [22XS_4[122X in the
  group  [22XO_8^+(2).2[122X,  and  the table of this wreath product can be constructed
  easily.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xtbl2:= CharacterTable("O8+(2).2");;[127X[104X
    [4X[25Xgap>[125X [27Xs3:= CharacterTable( "Symmetric", 3 );;[127X[104X
    [4X[25Xgap>[125X [27Xs:= CharacterTableWreathSymmetric( s3, 4 );[127X[104X
    [4X[28XCharacterTable( "Sym(3)wrS4" )[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  permutation character [10Xpi[110X of [22XO_8^+(2).2[122X can thus be computed by explicit
  induction, and the character of [22XO_8^+(2)[122X is obtained by restriction of [10Xpi[110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xfus:= PossibleClassFusions( s, tbl2 );[127X[104X
    [4X[28X[ [ 1, 41, 6, 3, 48, 9, 42, 19, 51, 8, 5, 50, 24, 49, 7, 2, 44, 22, [128X[104X
    [4X[28X      42, 12, 53, 17, 58, 21, 5, 47, 26, 50, 37, 52, 23, 60, 18, 4, [128X[104X
    [4X[28X      46, 25, 14, 61, 20, 9, 53, 30, 51, 26, 64, 8, 52, 31, 13, 56, [128X[104X
    [4X[28X      38 ] ][128X[104X
    [4X[25Xgap>[125X [27Xpi:= Induced( s, tbl2, [ TrivialCharacter( s ) ], fus[1] )[1];[127X[104X
    [4X[28XCharacter( CharacterTable( "O8+(2).2" ),[128X[104X
    [4X[28X [ 11200, 256, 160, 160, 80, 40, 40, 76, 13, 0, 0, 8, 8, 4, 0, 0, 16, [128X[104X
    [4X[28X  16, 4, 4, 4, 1, 1, 1, 1, 5, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 2, 2, 0, [128X[104X
    [4X[28X  0, 1120, 96, 0, 16, 0, 16, 8, 10, 4, 6, 7, 12, 3, 0, 0, 2, 0, 4, 0, [128X[104X
    [4X[28X  1, 1, 0, 0, 1, 0, 0, 0 ] )[128X[104X
    [4X[25Xgap>[125X [27XPermCharInfo( tbl2, pi ).ATLAS;[127X[104X
    [4X[28X[ "1a+84a+168a+175a+300a+700c+972a+1400a+3200a+4200b" ][128X[104X
    [4X[25Xgap>[125X [27Xtbl:= CharacterTable( "O8+(2)" );[127X[104X
    [4X[28XCharacterTable( "O8+(2)" )[128X[104X
    [4X[25Xgap>[125X [27Xrest:= RestrictedClassFunction( pi, tbl );[127X[104X
    [4X[28XCharacter( CharacterTable( "O8+(2)" ),[128X[104X
    [4X[28X [ 11200, 256, 160, 160, 160, 80, 40, 40, 40, 76, 13, 0, 0, 8, 8, 8, [128X[104X
    [4X[28X  4, 0, 0, 0, 16, 16, 16, 4, 4, 4, 4, 1, 1, 1, 1, 1, 1, 5, 0, 0, 0, [128X[104X
    [4X[28X  1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 2, 2, 2, 0, 0, 0 ] )[128X[104X
    [4X[25Xgap>[125X [27XPermCharInfo( tbl, rest ).ATLAS;[127X[104X
    [4X[28X[ "1a+84abc+175a+300a+700bcd+972a+3200a+4200a" ][128X[104X
  [4X[32X[104X
  
  
  [1X8.12 [33X[0;0YA Proof of Nonexistence of a Certain Subgroup[133X[101X
  
  [33X[0;0YWe  prove  that  the sporadic simple Mathieu group [22XG = M_22[122X (see [CCN+85, p.
  39])  has  no  subgroup  of index [22X56[122X. In [Isa76], remark after Theorem 5.18,
  this  is stated as an example of the case that a character may be a possible
  permutation  character  but not a permutation character. Let us consider the
  possible permutation character of degree [22X56[122X of [22XG[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xtbl:= CharacterTable( "M22" );[127X[104X
    [4X[28XCharacterTable( "M22" )[128X[104X
    [4X[25Xgap>[125X [27Xperms:= PermChars( tbl, rec( torso:= [ 56 ] ) );[127X[104X
    [4X[28X[ Character( CharacterTable( "M22" ),[128X[104X
    [4X[28X  [ 56, 8, 2, 4, 0, 1, 2, 0, 0, 2, 1, 1 ] ) ][128X[104X
    [4X[25Xgap>[125X [27Xpi:= perms[1];;[127X[104X
    [4X[25Xgap>[125X [27XNorm( pi );[127X[104X
    [4X[28X2[128X[104X
    [4X[25Xgap>[125X [27XDisplay( tbl, rec( chars:= perms ) );[127X[104X
    [4X[28XM22[128X[104X
    [4X[28X[128X[104X
    [4X[28X     2  7  7  2  5  4  .  2  .  .  3   .   .[128X[104X
    [4X[28X     3  2  1  2  .  .  .  1  .  .  .   .   .[128X[104X
    [4X[28X     5  1  .  .  .  .  1  .  .  .  .   .   .[128X[104X
    [4X[28X     7  1  .  .  .  .  .  .  1  1  .   .   .[128X[104X
    [4X[28X    11  1  .  .  .  .  .  .  .  .  .   1   1[128X[104X
    [4X[28X[128X[104X
    [4X[28X       1a 2a 3a 4a 4b 5a 6a 7a 7b 8a 11a 11b[128X[104X
    [4X[28X    2P 1a 1a 3a 2a 2a 5a 3a 7a 7b 4a 11b 11a[128X[104X
    [4X[28X    3P 1a 2a 1a 4a 4b 5a 2a 7b 7a 8a 11a 11b[128X[104X
    [4X[28X    5P 1a 2a 3a 4a 4b 1a 6a 7b 7a 8a 11a 11b[128X[104X
    [4X[28X    7P 1a 2a 3a 4a 4b 5a 6a 1a 1a 8a 11b 11a[128X[104X
    [4X[28X   11P 1a 2a 3a 4a 4b 5a 6a 7a 7b 8a  1a  1a[128X[104X
    [4X[28X[128X[104X
    [4X[28XY.1    56  8  2  4  .  1  2  .  .  2   1   1[128X[104X
  [4X[32X[104X
  
  [33X[0;0YSuppose  that [10Xpi[110X is a permutation character of [22XG[122X. Since [22XG[122X is [22X2[122X-transitive on
  the [22X56[122X cosets of the point stabilizer [22XS[122X, this stabilizer is transitive on [22X55[122X
  points,  and thus [22XG[122X has a subgroup [22XU[122X of index [22X56 ⋅ 55 = 3080[122X. We compute the
  possible permutation character of this degree.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xperms:= PermChars( tbl, rec( torso:= [ 56 * 55 ] ) );;[127X[104X
    [4X[25Xgap>[125X [27XLength( perms );[127X[104X
    [4X[28X16[128X[104X
  [4X[32X[104X
  
  [33X[0;0Y[22XU[122X is contained in [22XS[122X, so only those candidates must be considered that vanish
  on  all  classes where [10Xpi[110X vanishes. Furthermore, the index of [22XU[122X in [22XS[122X is odd,
  so  the  Sylow [22X2[122X subgroups of [22XU[122X and [22XS[122X are isomorphic; [22XS[122X contains elements of
  order [22X8[122X, hence also [22XU[122X does.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XOrdersClassRepresentatives( tbl );[127X[104X
    [4X[28X[ 1, 2, 3, 4, 4, 5, 6, 7, 7, 8, 11, 11 ][128X[104X
    [4X[25Xgap>[125X [27Xperms:= Filtered( perms, x -> x[5] = 0 and x[10] <> 0 );[127X[104X
    [4X[28X[ Character( CharacterTable( "M22" ),[128X[104X
    [4X[28X  [ 3080, 56, 2, 12, 0, 0, 2, 0, 0, 2, 0, 0 ] ), [128X[104X
    [4X[28X  Character( CharacterTable( "M22" ),[128X[104X
    [4X[28X  [ 3080, 8, 2, 8, 0, 0, 2, 0, 0, 4, 0, 0 ] ), [128X[104X
    [4X[28X  Character( CharacterTable( "M22" ),[128X[104X
    [4X[28X  [ 3080, 24, 11, 4, 0, 0, 3, 0, 0, 2, 0, 0 ] ), [128X[104X
    [4X[28X  Character( CharacterTable( "M22" ),[128X[104X
    [4X[28X  [ 3080, 24, 20, 4, 0, 0, 0, 0, 0, 2, 0, 0 ] ) ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YFor  getting  an  overview  of  the distribution of the elements of [22XU[122X to the
  conjugacy  classes  of  [22XG[122X,  we  use  the  output of [2XPermCharInfo[102X ([14XReference:
  PermCharInfo[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xinfoperms:= PermCharInfo( tbl, perms );;[127X[104X
    [4X[25Xgap>[125X [27XDisplay( tbl, infoperms.display );[127X[104X
    [4X[28XM22[128X[104X
    [4X[28X[128X[104X
    [4X[28X      2    7  7  2  5  2  3[128X[104X
    [4X[28X      3    2  1  2  .  1  .[128X[104X
    [4X[28X      5    1  .  .  .  .  .[128X[104X
    [4X[28X      7    1  .  .  .  .  .[128X[104X
    [4X[28X     11    1  .  .  .  .  .[128X[104X
    [4X[28X[128X[104X
    [4X[28X          1a 2a 3a 4a 6a 8a[128X[104X
    [4X[28X     2P   1a 1a 3a 2a 3a 4a[128X[104X
    [4X[28X     3P   1a 2a 1a 4a 2a 8a[128X[104X
    [4X[28X     5P   1a 2a 3a 4a 6a 8a[128X[104X
    [4X[28X     7P   1a 2a 3a 4a 6a 8a[128X[104X
    [4X[28X    11P   1a 2a 3a 4a 6a 8a[128X[104X
    [4X[28X[128X[104X
    [4X[28XI.1     3080 56  2 12  2  2[128X[104X
    [4X[28XI.2        1 21  8 54 24 36[128X[104X
    [4X[28XI.3        1  3  4  9 12 18[128X[104X
    [4X[28XI.4     3080  8  2  8  2  4[128X[104X
    [4X[28XI.5        1  3  8 36 24 72[128X[104X
    [4X[28XI.6        1  3  4  9 12 18[128X[104X
    [4X[28XI.7     3080 24 11  4  3  2[128X[104X
    [4X[28XI.8        1  9 44 18 36 36[128X[104X
    [4X[28XI.9        1  3  4  9 12 18[128X[104X
    [4X[28XI.10    3080 24 20  4  .  2[128X[104X
    [4X[28XI.11       1  9 80 18  . 36[128X[104X
    [4X[28XI.12       1  3  4  9 12 18[128X[104X
  [4X[32X[104X
  
  [33X[0;0YWe  have  four candidates. For each the above list shows first the character
  values,  then the cardinality of the intersection of [22XU[122X with the classes, and
  then  lower bounds for the lengths of [22XU[122X-conjugacy classes of these elements.
  Only  those  classes  of [22XG[122X are shown that contain elements of [22XU[122X for at least
  one of the characters.[133X
  
  [33X[0;0YIf  the  first  two candidates are permutation characters corresponding to [22XU[122X
  then  [22XU[122X contains exactly [22X8[122X elements of order [22X3[122X and thus [22XU[122X has a normal Sylow
  [22X3[122X  subgroup  [22XP[122X. But the order of [22XN_G(P)[122X is bounded by [22X72[122X, which can be shown
  as follows. The only elements in [22XG[122X with centralizer order divisible by [22X9[122X are
  of order [22X1[122X or [22X3[122X, so [22XP[122X is self-centralizing in [22XG[122X. The factor [22XN_G(P)/C_G(P)[122X is
  isomorphic  with a subgroup of Aut[22X(G) ≅ GL(2,3)[122X which has order divisible by
  [22X16[122X,  hence the order of [22XN_G(P)[122X divides [22X144[122X. Now note that [22X[ G : N_G(P) ] ≡ 1
  mod  3[122X by Sylow's Theorem, and [22X|G|/144 = 3080 ≡ -1 mod 3[122X. Thus the first two
  candidates are not permutation characters.[133X
  
  [33X[0;0YIf  the  last  two  candidates are permutation characters corresponding to [22XU[122X
  then  [22XU[122X has self-normalizing Sylow subgroups. This is because the index of a
  Sylow  [22X2[122X  normalizer  in [22XG[122X is odd and divides [22X9[122X, and if it is smaller than [22X9[122X
  then [22XU[122X contains at most [22X3 ⋅ 15 + 1[122X elements of [22X2[122X power order; the index of a
  Sylow [22X3[122X normalizer in [22XG[122X is congruent to [22X1[122X modulo [22X3[122X and divides [22X16[122X, and if it
  is smaller than [22X16[122X then [22XU[122X contains at most [22X4 ⋅ 8[122X elements of order [22X3[122X.[133X
  
  [33X[0;0YBut  since  [22XU[122X is solvable and not a [22Xp[122X-group, not all its Sylow subgroups can
  be self-normalizing; note that [22XU[122X has a proper normal subgroup [22XN[122X containing a
  Sylow  [22Xp[122X  subgroup  [22XP[122X  of [22XU[122X for a prime divisor [22Xp[122X of [22X|U|[122X, and [22XU = N ⋅ N_U(P)[122X
  holds by the Frattini argument (see [Hup67, Satz I.7.8]).[133X
  
  
  [1X8.13 [33X[0;0YA Permutation Character of the Lyons group[133X[101X
  
  [33X[0;0YLet  [22XG[122X be a maximal subgroup with structure [22X3^{2+4}:2A_5.D_8[122X in the sporadic
  simple  Lyons group [22XLy[122X. We want to compute the permutation character [22X1_G^Ly[122X.
  (This  construction  has  been  explained  in [BP98,  Section  4.2], without
  showing explicit [5XGAP[105X code.)[133X
  
  [33X[0;0YIn the representation of [22XLy[122X as automorphism group of the rank [22X5[122X graph [10XB[110X with
  [22X9606125[122X  points  (see [CCN+85,  p.  174]), [22XG[122X is the stabilizer of an edge. A
  group  [22XS[122X  with  structure  [22X3.McL.2[122X is the point stabilizer. So the two point
  stabilizer  [22XU  = S ∩ G[122X is a subgroup of index [22X2[122X in [22XG[122X. The index of [22XU[122X in [22XS[122X is
  [22X15400[122X, and according to the list of maximal subgroups of [22XMcL.2[122X (see [CCN+85,
  p. 100]), the group [22XU[122X is isomorphic to the preimage in [22X3.McL.2[122X of a subgroup
  [22XH[122X of [22XMcL.2[122X with structure [22X3_+^{1+4}:4S_5[122X.[133X
  
  [33X[0;0YUsing  the  improved  combinatorial method described in [BP98, Section 3.2],
  all  possible  permutation  characters of degree [22X15400[122X for the group [22XMcL[122X are
  computed. (The method of [BP98, Section 3.3] is slower but also needs only a
  few seconds.)[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xly:= CharacterTable( "Ly" );;[127X[104X
    [4X[25Xgap>[125X [27Xmcl:= CharacterTable( "McL" );;[127X[104X
    [4X[25Xgap>[125X [27Xmcl2:= CharacterTable( "McL.2" );;[127X[104X
    [4X[25Xgap>[125X [27X3mcl2:= CharacterTable( "3.McL.2" );;[127X[104X
    [4X[25Xgap>[125X [27Xperms:= PermChars( mcl, rec( degree:= 15400 ) );[127X[104X
    [4X[28X[ Character( CharacterTable( "McL" ),[128X[104X
    [4X[28X  [ 15400, 56, 91, 10, 12, 25, 0, 11, 2, 0, 0, 2, 1, 1, 1, 0, 0, 3, [128X[104X
    [4X[28X      0, 0, 1, 1, 1, 1 ] ), Character( CharacterTable( "McL" ),[128X[104X
    [4X[28X  [ 15400, 280, 10, 37, 20, 0, 5, 10, 1, 0, 0, 2, 1, 1, 0, 0, 0, 2, [128X[104X
    [4X[28X      0, 0, 0, 0, 0, 0 ] ) ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YWe get two characters, corresponding to the two classes of maximal subgroups
  of  index [22X15400[122X in [22XMcL[122X. The permutation character [22Xπ = 1_{H ∩ McL}^McL[122X is the
  one  with  nonzero  value  on the class [10X10A[110X, since the subgroup of structure
  [22X2S_5[122X in [22XH ∩ McL[122X contains elements of order [22X10[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xord10:= Filtered( [ 1 .. NrConjugacyClasses( mcl ) ],[127X[104X
    [4X[25X>[125X [27X                     i -> OrdersClassRepresentatives( mcl )[i] = 10 );[127X[104X
    [4X[28X[ 15 ][128X[104X
    [4X[25Xgap>[125X [27XList( perms, pi -> pi[ ord10[1] ] );[127X[104X
    [4X[28X[ 1, 0 ][128X[104X
    [4X[25Xgap>[125X [27Xpi:= perms[1];[127X[104X
    [4X[28XCharacter( CharacterTable( "McL" ),[128X[104X
    [4X[28X [ 15400, 56, 91, 10, 12, 25, 0, 11, 2, 0, 0, 2, 1, 1, 1, 0, 0, 3, 0, [128X[104X
    [4X[28X  0, 1, 1, 1, 1 ] )[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  character  [22X1_H^McL.2[122X  is  an  extension  of [22Xπ[122X, so we can use the method
  of [BP98,  Section  3.3]  to compute all possible permutation characters for
  the  group  [22XMcL.2[122X  that  have the values of [22Xπ[122X on the classes of [22XMcL[122X. We find
  that  the  extension  of  [22Xπ[122X  to  a permutation character of [22XMcL.2[122X is unique.
  Regarded as a character of [22X3.McL.2[122X, this character is equal to [22X1_U^S[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xmap:= InverseMap( GetFusionMap( mcl, mcl2 ) );[127X[104X
    [4X[28X[ 1, 2, 3, 4, 5, 6, 7, 8, 9, [ 10, 11 ], 12, [ 13, 14 ], 15, 16, 17, [128X[104X
    [4X[28X  18, [ 19, 20 ], [ 21, 22 ], [ 23, 24 ] ][128X[104X
    [4X[25Xgap>[125X [27Xtorso:= CompositionMaps( pi, map );[127X[104X
    [4X[28X[ 15400, 56, 91, 10, 12, 25, 0, 11, 2, 0, 2, 1, 1, 0, 0, 3, 0, 1, 1 ][128X[104X
    [4X[25Xgap>[125X [27Xperms:= PermChars( mcl2, rec( torso:= torso ) );[127X[104X
    [4X[28X[ Character( CharacterTable( "McL.2" ),[128X[104X
    [4X[28X  [ 15400, 56, 91, 10, 12, 25, 0, 11, 2, 0, 2, 1, 1, 0, 0, 3, 0, 1, [128X[104X
    [4X[28X      1, 110, 26, 2, 4, 0, 0, 5, 2, 1, 1, 0, 0, 1, 1 ] ) ][128X[104X
    [4X[25Xgap>[125X [27Xpi:= Inflated( perms[1], 3mcl2 );[127X[104X
    [4X[28XCharacter( CharacterTable( "3.McL.2" ),[128X[104X
    [4X[28X [ 15400, 15400, 56, 56, 91, 91, 10, 12, 12, 25, 25, 0, 0, 11, 11, 2, [128X[104X
    [4X[28X  2, 0, 0, 0, 2, 2, 1, 1, 1, 0, 0, 0, 0, 3, 3, 0, 0, 0, 1, 1, 1, 1, [128X[104X
    [4X[28X  1, 1, 110, 26, 2, 4, 0, 0, 5, 2, 1, 1, 0, 0, 1, 1 ] )[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  fusion  of  conjugacy  classes  of  [22XS[122X  in  [22XLy[122X  can be computed from the
  character  tables  of  [22XS[122X and [22XLy[122X given in [CCN+85], it is unique up to Galois
  automorphisms of the table of [22XLy[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xfus:= PossibleClassFusions( 3mcl2, ly );;  Length( fus );[127X[104X
    [4X[28X4[128X[104X
    [4X[25Xgap>[125X [27Xg:= AutomorphismsOfTable( ly );;[127X[104X
    [4X[25Xgap>[125X [27XOrbitLengths( g, fus, OnTuples );    [127X[104X
    [4X[28X[ 4 ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YNow we can induce [22X1_U^S[122X to [22XLy[122X, which yields [22X(1_U^S)^Ly = 1_U^Ly[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xpi:= Induced( 3mcl2, ly, [ pi ], fus[1] )[1];[127X[104X
    [4X[28XCharacter( CharacterTable( "Ly" ),[128X[104X
    [4X[28X [ 147934325000, 286440, 1416800, 1082, 784, 12500, 0, 672, 42, 24, [128X[104X
    [4X[28X  0, 40, 0, 2, 20, 0, 0, 0, 64, 10, 0, 50, 2, 0, 0, 4, 0, 0, 0, 0, 4, [128X[104X
    [4X[28X  0, 0, 0, 0, 2, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ] )[128X[104X
  [4X[32X[104X
  
  [33X[0;0YAll  elements of odd order in [22XG[122X are contained in [22XU[122X, for such an element [22Xg[122X we
  have[133X
  
  
  [24X[33X[0;6Y1_G^Ly(g) = |C_Ly(g)| / |G| ⋅ |G ∩ Cl_Ly(g)| = |C_Ly(g)| / (2 ⋅ |U|) ⋅ |U ∩ Cl_Ly(g)| = 1/2 ⋅ 1_U^Ly(g) ,[133X[124X
  
  [33X[0;0Yso  we  can  prescribe  the  values  of [22X1_G^Ly[122X on all classes of odd element
  order. For elements [22Xg[122X of even order we have the weaker condition [22XU∩ Cl_Ly(g)
  ⊆  G  ∩  Cl_Ly(g)[122X  and  thus  [22X1_G^Ly(g) ≥ 1/2 ⋅ 1_U^Ly(g)[122X, which gives lower
  bounds for the value of [22X1_G^Ly[122X on the remaining classes.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xorders:= OrdersClassRepresentatives( ly );[127X[104X
    [4X[28X[ 1, 2, 3, 3, 4, 5, 5, 6, 6, 6, 7, 8, 8, 9, 10, 10, 11, 11, 12, 12, [128X[104X
    [4X[28X  14, 15, 15, 15, 18, 20, 21, 21, 22, 22, 24, 24, 24, 25, 28, 30, 30, [128X[104X
    [4X[28X  31, 31, 31, 31, 31, 33, 33, 37, 37, 40, 40, 42, 42, 67, 67, 67 ][128X[104X
    [4X[25Xgap>[125X [27Xtorso:= [];;                                   [127X[104X
    [4X[25Xgap>[125X [27Xfor i in [ 1 .. Length( orders ) ] do[127X[104X
    [4X[25X>[125X [27X     if orders[i] mod 2 = 1 then[127X[104X
    [4X[25X>[125X [27X       torso[i]:= pi[i]/2;[127X[104X
    [4X[25X>[125X [27X     fi;[127X[104X
    [4X[25X>[125X [27X   od;[127X[104X
    [4X[25Xgap>[125X [27Xtorso;[127X[104X
    [4X[28X[ 73967162500,, 708400, 541,, 6250, 0,,,, 0,,, 1,,, 0, 0,,,, 25, 1, 0,[128X[104X
    [4X[28X  ,, 0, 0,,,,,, 0,,,, 0, 0, 0, 0, 0, 0, 0, 0, 0,,,,, 0, 0, 0 ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YExactly one possible permutation character of [22XLy[122X satisfies these conditions.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xperms:= PermChars( ly, rec( torso:= torso ) );;[127X[104X
    [4X[25Xgap>[125X [27XLength( perms );[127X[104X
    [4X[28X43[128X[104X
    [4X[25Xgap>[125X [27Xperms:= Filtered( perms, cand -> ForAll( [ 1 .. Length( orders ) ],[127X[104X
    [4X[25X>[125X [27X       i -> cand[i] >= pi[i] / 2 ) );[127X[104X
    [4X[28X[ Character( CharacterTable( "Ly" ),[128X[104X
    [4X[28X  [ 73967162500, 204820, 708400, 541, 392, 6250, 0, 1456, 61, 25, 0, [128X[104X
    [4X[28X      22, 10, 1, 10, 0, 0, 0, 32, 5, 0, 25, 1, 0, 1, 2, 0, 0, 0, 0, [128X[104X
    [4X[28X      4, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 0, 0, 0, [128X[104X
    [4X[28X      0, 0 ] ) ][128X[104X
  [4X[32X[104X
  
  [33X[0;0Y(The  permutation  character [22X1_G^Ly[122X was used in the proof that the character
  [22Xχ_37[122X  of  [22XLy[122X  (see [CCN+85,  p. 175]) occurs with multiplicity at least 2 in
  each character of [22XLy[122X that is induced from a proper subgroup of [22XLy[122X.)[133X
  
  
  [1X8.14 [33X[0;0YIdentifying two subgroups of Aut[22X(U_3(5))[122X[101X[1X (October 2001)[133X[101X
  
  [33X[0;0YAccording  to  the  Atlas  of  Finite  Groups [CCN+85,  p.  34],  the  group
  Aut[22X(U_3(5))[122X  has  two classes of maximal subgroups of order [22X2^4 ⋅ 3^3[122X, which
  have the structures [22X3^2 : 2S_4[122X and [22X6^2 : D_12[122X, respectively.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xtbl:= CharacterTable( "U3(5).3.2" );[127X[104X
    [4X[28XCharacterTable( "U3(5).3.2" )[128X[104X
    [4X[25Xgap>[125X [27Xdeg:= Size( tbl ) / ( 2^4*3^3 );[127X[104X
    [4X[28X1750[128X[104X
    [4X[25Xgap>[125X [27Xpi:= PermChars( tbl, rec( torso:= [ deg ] ) );[127X[104X
    [4X[28X[ Character( CharacterTable( "U3(5).3.2" ),[128X[104X
    [4X[28X  [ 1750, 70, 13, 2, 0, 0, 1, 0, 0, 0, 10, 7, 10, 4, 2, 0, 0, 0, 0, [128X[104X
    [4X[28X      0, 0, 30, 10, 3, 0, 0, 1, 0, 0 ] ), [128X[104X
    [4X[28X  Character( CharacterTable( "U3(5).3.2" ),[128X[104X
    [4X[28X  [ 1750, 30, 4, 6, 0, 0, 0, 0, 0, 0, 40, 7, 0, 6, 0, 0, 0, 0, 0, 0, [128X[104X
    [4X[28X      0, 20, 0, 2, 2, 0, 0, 0, 0 ] ) ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YNow  the  question is which character belongs to which subgroup. We see that
  the  first  character  vanishes  on  the  classes of element order [22X8[122X and the
  second  does  not,  so  only  the first one can be the permutation character
  induced from [22X6^2 : D_12[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xord8:= Filtered( [ 1 .. NrConjugacyClasses( tbl ) ],[127X[104X
    [4X[25X>[125X [27X              i -> OrdersClassRepresentatives( tbl )[i] = 8 );[127X[104X
    [4X[28X[ 9, 25 ][128X[104X
    [4X[25Xgap>[125X [27XList( pi, x -> x{ ord8 } );[127X[104X
    [4X[28X[ [ 0, 0 ], [ 0, 2 ] ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YThus  the  question  is whether the second candidate is really a permutation
  character.  Since  none of the two candidates vanishes on any outer coset of
  [22XU_3(5)[122X  in  Aut[22X(U_3(5))[122X,  the  point stabilizers are extensions of groups of
  order  [22X2^3 ⋅ 3^2[122X in [22XU_3(5)[122X. The restrictions of the candidates to [22XU_3(5)[122X are
  different, so we can try to answer the question using information about this
  group.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xsubtbl:= CharacterTable( "U3(5)" );[127X[104X
    [4X[28XCharacterTable( "U3(5)" )[128X[104X
    [4X[25Xgap>[125X [27Xrest:= RestrictedClassFunctions( pi, subtbl );[127X[104X
    [4X[28X[ Character( CharacterTable( "U3(5)" ),[128X[104X
    [4X[28X  [ 1750, 70, 13, 2, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0 ] ), [128X[104X
    [4X[28X  Character( CharacterTable( "U3(5)" ),[128X[104X
    [4X[28X  [ 1750, 30, 4, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ] ) ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  intersection  of  the  [22X3^2  : 2S_4[122X subgroup with [22XU_3(5)[122X lies inside the
  maximal  subgroup  of  type [22XM_10[122X, which does not contain elements of order[22X6[122X.
  Only the second character has this property.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xord6:= Filtered( [ 1 .. NrConjugacyClasses( subtbl ) ],[127X[104X
    [4X[25X>[125X [27X              i -> OrdersClassRepresentatives( subtbl )[i] = 6 );[127X[104X
    [4X[28X[ 9 ][128X[104X
    [4X[25Xgap>[125X [27XList( rest, x -> x{ ord6 } );[127X[104X
    [4X[28X[ [ 1 ], [ 0 ] ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YIn order to establish the two characters as permutation characters, we could
  also  compute  the permutation characters of the degree in question directly
  from  the table of marks of [22XU_3(5)[122X, which is contained in the [5XGAP[105X library of
  tables of marks.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xtom:= TableOfMarks( "U3(5)" );[127X[104X
    [4X[28XTableOfMarks( "U3(5)" )[128X[104X
    [4X[25Xgap>[125X [27Xperms:= PermCharsTom( subtbl, tom );;[127X[104X
    [4X[25Xgap>[125X [27XSet( Filtered( perms, x -> x[1] = deg ) ) = Set( rest );[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [33X[0;0YWe were mainly interested in the multiplicities of irreducible characters in
  these characters. The action of Aut[22X(U_3(5)[122X on the cosets of [22X3^2 : 2S_4[122X turns
  out to be multiplicity-free whereas that on the cosets of [22X6^2 : D_12[122X is not.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XPermCharInfo( tbl, pi ).ATLAS;[127X[104X
    [4X[28X[ "1a+21a+42a+84aac+105a+125a+126a+250a+252a+288bc", [128X[104X
    [4X[28X  "1a+42a+84ac+105ab+125a+126a+250a+252b+288bc" ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YIt  should be noted that the restrictions of the multiplicity-free character
  to   the   subgroups   [22XU_3(5).2[122X   and   [22XU_3(5).3[122X   of   Aut[22X(U_3(5)[122X  are  not
  multiplicity-free.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xsubtbl2:= CharacterTable( "U3(5).2" );;[127X[104X
    [4X[25Xgap>[125X [27Xrest2:= RestrictedClassFunctions( pi, subtbl2 );;[127X[104X
    [4X[25Xgap>[125X [27XPermCharInfo( subtbl2, rest2 ).ATLAS;[127X[104X
    [4X[28X[ "1a+21aab+28aa+56aa+84a+105a+125aab+126aab+288aa", [128X[104X
    [4X[28X  "1a+21ab+28a+56a+84a+105ab+125aab+126a+252a+288aa" ][128X[104X
    [4X[25Xgap>[125X [27Xsubtbl3:= CharacterTable( "U3(5).3" );;[127X[104X
    [4X[25Xgap>[125X [27Xrest3:= RestrictedClassFunctions( pi, subtbl3 );;[127X[104X
    [4X[25Xgap>[125X [27XPermCharInfo( subtbl3, rest3 ).ATLAS;[127X[104X
    [4X[28X[ "1a+21abc+84aab+105a+125abc+126abc+144bcef", [128X[104X
    [4X[28X  "1a+21bc+84ab+105aa+125abc+126adg+144bcef" ][128X[104X
  [4X[32X[104X
  
  
  [1X8.15 [33X[0;0YA Permutation Character of Aut[22X(O_8^+(2))[122X[101X[1X (October 2001)[133X[101X
  
  [33X[0;0YAccording  to  the  Atlas  of  Finite  Groups [CCN+85, p. 85], the group [22XG =[122X
  Aut[22X(O_8^+(2))[122X has a class of maximal subgroups of order [22X2^13 ⋅ 3^2[122X, thus the
  index  of  these  subgroups in [22XG[122X is [22X3^4 ⋅ 5^2 ⋅ 7[122X. The intersection of these
  subgroups  with [22XH = O_8^+(2)[122X lie inside maximal subgroups of type [22X2^6 : A_8[122X.
  We  want  to  show  that the permutation character of the action of [22XG[122X on the
  cosets of these subgroups is not multiplicity-free.[133X
  
  [33X[0;0YSince  the  table of marks for [22XH[122X is available in [5XGAP[105X, but not that for [22XG[122X, we
  first  compute  the  [22XH[122X-permutation characters of the intersections with [22XH[122X of
  index [22X3^4 ⋅ 5^2 ⋅ 7 = 14175[122X subgroups in [22XG[122X.[133X
  
  [33X[0;0Y(Note  that  these  intersections  have  order [22X2^12 ⋅ 3[122X because subgroups of
  order  [22X2^12  ⋅  3^2[122X are contained in [22XO_8^+(2).2[122X and hence are not maximal in
  [22XG[122X.)[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xt:= CharacterTable( "O8+(2).3.2" );;[127X[104X
    [4X[25Xgap>[125X [27Xs:= CharacterTable( "O8+(2)" );;[127X[104X
    [4X[25Xgap>[125X [27Xtom:= TableOfMarks( s );;[127X[104X
    [4X[25Xgap>[125X [27Xperms:= PermCharsTom( s, tom );;[127X[104X
    [4X[25Xgap>[125X [27Xdeg:= 3^4*5^2*7;[127X[104X
    [4X[28X14175[128X[104X
    [4X[25Xgap>[125X [27Xperms:= Filtered( perms, x -> x[1] = deg );;[127X[104X
    [4X[25Xgap>[125X [27XLength( perms );[127X[104X
    [4X[28X4[128X[104X
    [4X[25Xgap>[125X [27XLength( Set( perms ) );[127X[104X
    [4X[28X1[128X[104X
  [4X[32X[104X
  
  [33X[0;0YWe  see  that  there are four classes of subgroups [22XS[122X in [22XH[122X that may belong to
  maximal  subgroups  of  the  desired  index  in  [22XG[122X, and that the permutation
  characters  are  equal.  They lead to such groups if they extend to [22XG[122X, so we
  compute   the  possible  permutation  characters  of  [22XG[122X  that  extend  these
  characters.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xfus:= PossibleClassFusions( s, t );[127X[104X
    [4X[28X[ [ 1, 2, 3, 3, 3, 4, 5, 5, 5, 6, 7, 8, 9, 10, 10, 10, 11, 12, 12, [128X[104X
    [4X[28X      12, 13, 13, 13, 14, 14, 14, 15, 16, 16, 16, 17, 17, 17, 18, 19, [128X[104X
    [4X[28X      20, 21, 22, 22, 22, 23, 23, 23, 24, 24, 24, 25, 26, 26, 26, 27, [128X[104X
    [4X[28X      27, 27 ] ][128X[104X
    [4X[25Xgap>[125X [27Xfus:= fus[1];;[127X[104X
    [4X[25Xgap>[125X [27Xinv:= InverseMap( fus );;[127X[104X
    [4X[25Xgap>[125X [27Xcomp:= CompositionMaps( perms[1], inv );[127X[104X
    [4X[28X[ 14175, 1215, 375, 79, 0, 0, 27, 27, 99, 15, 7, 0, 0, 0, 0, 9, 3, 1, [128X[104X
    [4X[28X  0, 1, 1, 0, 0, 0, 0, 0, 0 ][128X[104X
    [4X[25Xgap>[125X [27Xext:= PermChars( t, rec( torso:= comp ) );[127X[104X
    [4X[28X[ Character( CharacterTable( "O8+(2).3.2" ),[128X[104X
    [4X[28X  [ 14175, 1215, 375, 79, 0, 0, 27, 27, 99, 15, 7, 0, 0, 0, 0, 9, 3, [128X[104X
    [4X[28X      1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 63, 9, 15, 7, 1, 0, 3, 3, 3, 1, [128X[104X
    [4X[28X      0, 0, 1, 1, 945, 129, 45, 69, 21, 25, 13, 0, 0, 0, 9, 0, 3, 3, [128X[104X
    [4X[28X      7, 1, 0, 0, 0, 3, 1, 0, 0, 0, 0, 0, 0 ] ) ][128X[104X
    [4X[25Xgap>[125X [27XPermCharInfo( t, ext[1] ).ATLAS;[127X[104X
    [4X[28X[ "1a+50b+100a+252bb+300b+700b+972bb+1400a+1944a+3200b+4032b" ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YThus we get one permutation character of [22XG[122X which is not multiplicity-free.[133X
  
  
  [1X8.16 [33X[0;0YFour Primitive Permutation Characters of the Monster Group[133X[101X
  
  [33X[0;0YIn  this  section, we compute four primitive permutation characters [22X1_H^M[122X of
  the sporadic simple Monster group [22XM[122X, using the following strategy.[133X
  
  [33X[0;0YLet  [22XE[122X  be  an  elementary  abelian  [22X2[122X-subgroup of [22XM[122X, and [22XH = N_M(E)[122X. For an
  involution  [22Xz  ∈  E[122X, let [22XG = C_M(z)[122X and [22XU = G ∩ H = C_H(z)[122X and [22XV = C_H(E)[122X, a
  normal  subgroup  of  [22XH[122X. According to the Atlas of Finite Groups [CCN+85, p.
  234],  [22XG[122X  has  the structure [22X2.B[122X if [22Xz[122X is in the class [10X2A[110X of [22XM[122X, and [22XG[122X has the
  structure  [22X2^{1+24}_+.Co_1[122X if [22Xz[122X is in the class [10X2B[110X of [22XM[122X. In the latter case,
  let  [22XN[122X  denote  the extraspecial normal subgroup of order [22X2^25[122X in [22XG[122X. It will
  turn out that in our situation, [22XU[122X contains [22XN[122X.[133X
  
  [33X[0;0YWe  want  to  compute many values of [22X1_H^M[122X from the knowledge of permutation
  characters  [22X1_X^M[122X, for suitable subgroups [22XX[122X with the property [22XV ≤ X ≤ U[122X, and
  then use the [5XGAP[105X function [2XPermChars[102X ([14XReference: PermChars[114X) for computing all
  those  possible  permutation  characters of [22XM[122X that take the known values; if
  there is a unique solution then this is the desired character [22X1_H^M[122X.[133X
  
                       M
                      ╱ ╲
                     G   ╲
                      ╲   H
                       ╲ ╱
                        U
                         ╲
                          VN
                         ╱ ╲
                        V   N
                         ╲ ╱ 
                         V∩N
                          │
                          Z
                          │
                          1
  
  [33X[0;0YWhy  does  this approach have a chance to be successful? Currently we do not
  have  representations  for  the  subgroups  [22XH[122X in question, but the character
  tables  of the involution centralizers [22XG[122X in [22XM[122X are available, and also either
  the  character tables of [22XX/V[122X for the interesting subgroups [22XX[122X are known or we
  have enough information to compute the characters [22X1_X^G[122X.[133X
  
  [33X[0;0YAnd  how do we compute certain values of [22X1_H^M[122X? Suppose that [22XC[122X is a union of
  classes  of [22XM[122X and [22XI[122X is an index set such that [22X(1_H)_{C ∩ H} = (∑_{i ∈ I} c_i
  1_{X_i}^H)_{C ∩ H}[122X holds for suitable rational numbers [22Xc_i[122X.[133X
  
  [33X[0;0YThe  right  hand side of this equality lives in [22XH/V[122X, provided that [22XC[122X [21Xbehaves
  well[121X  w.r.t.  factoring out the normal subgroup [22XV[122X of [22XH[122X, i. e., if there is a
  set  of classes in [22XH/V[122X whose preimages in [22XH[122X form the set [22XH ∩ C[122X. For example,
  [22XC[122X  may be the set of all those elements in [22XM[122X whose order is not divisible by
  a particular prime [22Xp[122X that divides [22X|H|[122X but not [22X|U|[122X.[133X
  
  [33X[0;0YUnder these conditions, we have [22X(1_H^M)_C = ((∑_{i ∈ I} c_i 1_{X_i}^G)^M)_C[122X,
  and  we  interpret  the  right  hand side as follows: If [22XX_i[122X contains [22XN[122X then
  [22X1_{X_i}^G[122X can be identified with [22X1_{X_i/N}^{G/N}[122X. If [22XX_i[122X contains at least [22XZ[122X
  then  [22X1_{X_i}^G[122X  can be identified with [22X1_{X_i/Z}^{G/Z}[122X. As mentioned above,
  we  have  good chances to compute these characters. So the main task in each
  of  the  following  sections  is to find, for a suitable set [22XC[122X of classes, a
  linear  combination of permutation characters of [22XH/V[122X whose restriction to [22X(C
  ∩ H) / V[122X is constant and nonzero.[133X
  
  
  [1X8.16-1 [33X[0;0YThe Subgroup [22X2^2.2^11.2^22.(S_3 × M_24)[122X[101X[1X (June 2009)[133X[101X
  
  [33X[0;0YAccording  to the Atlas of Finite Groups [CCN+85, p. 234], the Monster group
  [22XM[122X has a class of maximal subgroups [22XH[122X of the type [22X2^2.2^11.2^22.(S_3 × M_24)[122X.
  Currently  the  character  table  of  [22XH[122X  and the class fusion into [22XM[122X are not
  available  in [5XGAP[105X. We are interested in the permutation character [22X1_H^G[122X, and
  we will compute it without this information.[133X
  
  [33X[0;0YThe  subgroup [22XH[122X normalizes a Klein four group [22XE[122X whose involutions lie in the
  class [10X2B[110X. We fix an involution [22Xz[122X in [22XE[122X, and set [22XG = C_M(z)[122X, [22XU = C_H(z)[122X, and [22XV
  =  C_H(E)[122X.  Further, let [22XN[122X be the extraspecial normal subgroup of order [22X2^25[122X
  in [22XG[122X.[133X
  
  [33X[0;0YSo  [22XG[122X  has  the  structure  [22X2^{1+24}_+.Co_1[122X, and [22XU[122X has index three in [22XH[122X. The
  order  of  [22XN U / N[122X is a multiple of [22X2^{2+11+22-25} ⋅ 2 ⋅ |M_24|[122X, and [22XN U / N[122X
  occurs as a subgroup of [22XG / N ≅ Co_1[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xco1:= CharacterTable( "Co1" );;[127X[104X
    [4X[25Xgap>[125X [27Xorder:= 2^(2+11+22-25) * 2 * Size( CharacterTable( "M24" ) );[127X[104X
    [4X[28X501397585920[128X[104X
    [4X[25Xgap>[125X [27Xmaxes:= List( Maxes( co1 ), CharacterTable );;[127X[104X
    [4X[25Xgap>[125X [27Xfilt:= Filtered( maxes, t -> Size( t ) mod order = 0 );[127X[104X
    [4X[28X[ CharacterTable( "2^11:M24" ) ][128X[104X
    [4X[25Xgap>[125X [27XList( filt, t -> Size( t ) / order );[127X[104X
    [4X[28X[ 1 ][128X[104X
    [4X[25Xgap>[125X [27Xk:= filt[1];;[127X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  list  of maximal subgroups of [22XCo_1[122X (see [CCN+85, p. 183]) tells us that
  [22XNU  /  N[122X is a maximal subgroup [22XK[122X of [22XCo_1[122X and has the structure [22X2^11:M_24[122X. In
  particular, [22XU[122X contains [22XN[122X and thus [22XU/N ≅ K[122X.[133X
  
  [33X[0;0YLet [22XC = { g ∈ M; 3 ∤ |g|[122X or [22X1_V^M(g^3) = 0 }[122X.[133X
  
  [33X[0;0YThen  [22X(1_H)_{C  ∩ H} = (1_U^H - 1/3 1_V^H)_{C ∩ H}[122X holds, as we can see from
  computations with [22XH/V ≅ S_3[122X, as follows.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xf:= CharacterTable( "Symmetric", 3 );[127X[104X
    [4X[28XCharacterTable( "Sym(3)" )[128X[104X
    [4X[25Xgap>[125X [27XOrdersClassRepresentatives( f );[127X[104X
    [4X[28X[ 1, 2, 3 ][128X[104X
    [4X[25Xgap>[125X [27Xdeg3:= PermChars( f, 3 );[127X[104X
    [4X[28X[ Character( CharacterTable( "Sym(3)" ), [ 3, 1, 0 ] ) ][128X[104X
    [4X[25Xgap>[125X [27Xdeg6:= PermChars( f, 6 );[127X[104X
    [4X[28X[ Character( CharacterTable( "Sym(3)" ), [ 6, 0, 0 ] ) ][128X[104X
    [4X[25Xgap>[125X [27Xdeg3[1] - 1/3 * deg6[1];[127X[104X
    [4X[28XClassFunction( CharacterTable( "Sym(3)" ), [ 1, 1, 0 ] )[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  character  table  of  [22XG[122X  is  available  in  [5XGAP[105X,  so we can compute the
  permutation  character  [22Xπ  =  1_U^G[122X  by  computing the primitive permutation
  character  [22X1_K^{Co_1}[122X, identifying it with [22X1_{U/N}^{G/N}[122X, and then inflating
  this character to [22XG[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xm:= CharacterTable( "M" );[127X[104X
    [4X[28XCharacterTable( "M" )[128X[104X
    [4X[25Xgap>[125X [27Xg:= CharacterTable( "MC2B" );[127X[104X
    [4X[28XCharacterTable( "2^1+24.Co1" )[128X[104X
    [4X[25Xgap>[125X [27Xpi:= RestrictedClassFunction( TrivialCharacter( k )^co1, g );;[127X[104X
  [4X[32X[104X
  
  [33X[0;0YNext  we  consider the permutation character [22Xϕ = 1_V^G[122X. The group [22XV[122X does not
  contain  [22XN[122X  because [22XK[122X is perfect. But [22XV[122X contains [22XZ[122X because otherwise [22XU[122X would
  be  a  direct product of [22XV[122X and [22XZ[122X, which would imply that [22XN[122X would be a direct
  product of [22XV ∩ N[122X and [22XZ[122X. So we can regard [22Xϕ[122X as the inflation of [22X1_{V/Z}^{G/Z}[122X
  from  [22XG/Z[122X  to  [22XG[122X,  i. e., we can perform the computations with the character
  table of the factor group [22XG/Z[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xzclasses:= ClassPositionsOfCentre( g );;[127X[104X
    [4X[25Xgap>[125X [27Xgmodz:= g / zclasses;[127X[104X
    [4X[28XCharacterTable( "2^1+24.Co1/[ 1, 2 ]" )[128X[104X
    [4X[25Xgap>[125X [27Xinvmap:= InverseMap( GetFusionMap( g, gmodz ) );;[127X[104X
    [4X[25Xgap>[125X [27Xpibar:= CompositionMaps( pi, invmap );;[127X[104X
  [4X[32X[104X
  
  [33X[0;0YSince  [22Xϕ(g) = [G:V] ⋅ |g^G ∩ V| / |g^G|[122X holds for [22Xg ∈ G[122X, and since [22Xg^G ∩ V ⊆
  g^G  ∩ VN[122X, with equality if [22Xg[122X has odd order, we get [22Xϕ(g) = 2 ⋅ π(g)[122X if [22Xg[122X has
  odd order, and [22Xϕ(g) = 0[122X if [22Xπ(g) = 0[122X.[133X
  
  [33X[0;0YWe want to compute the possible permutation characters with these values.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xfactorders:= OrdersClassRepresentatives( gmodz );;[127X[104X
    [4X[25Xgap>[125X [27Xphibar:= [];;[127X[104X
    [4X[25Xgap>[125X [27Xfor i in [ 1 .. NrConjugacyClasses( gmodz ) ] do[127X[104X
    [4X[25X>[125X [27X     if factorders[i] mod 2 = 1 then[127X[104X
    [4X[25X>[125X [27X       phibar[i]:= 2 * pibar[i];[127X[104X
    [4X[25X>[125X [27X     elif pibar[i] = 0 then[127X[104X
    [4X[25X>[125X [27X       phibar[i]:= 0;[127X[104X
    [4X[25X>[125X [27X     fi;[127X[104X
    [4X[25X>[125X [27X   od;[127X[104X
    [4X[25Xgap>[125X [27Xcand:= PermChars( gmodz, rec( torso:= phibar ) );;[127X[104X
    [4X[25Xgap>[125X [27XLength( cand );[127X[104X
    [4X[28X1[128X[104X
  [4X[32X[104X
  
  [33X[0;0YNow we know [22Xπ^M = 1_U^M[122X and [22Xϕ^M = 1_V^M[122X, so we can write down [22X(1_H^M)_C[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xphi:= RestrictedClassFunction( cand[1], g )^m;;[127X[104X
    [4X[25Xgap>[125X [27Xpi:= pi^m;;[127X[104X
    [4X[25Xgap>[125X [27Xcand:= ShallowCopy( pi - 1/3 * phi );;[127X[104X
    [4X[25Xgap>[125X [27Xmorders:= OrdersClassRepresentatives( m );;[127X[104X
    [4X[25Xgap>[125X [27Xfor i in [ 1 .. Length( morders ) ] do[127X[104X
    [4X[25X>[125X [27X     if morders[i] mod 3 = 0 and phi[ PowerMap( m, 3 )[i] ] <> 0 then[127X[104X
    [4X[25X>[125X [27X       Unbind( cand[i] );[127X[104X
    [4X[25X>[125X [27X     fi;[127X[104X
    [4X[25X>[125X [27X   od;[127X[104X
  [4X[32X[104X
  
  [33X[0;0YWe  claim that [22X1_H^M(g) ≥ π^M(g) - 1/3 ψ^M(g)[122X for all [22Xg ∈ M[122X. In order to see
  this,  let  [22XH'[122X denote the index two subgroup of [22XH[122X, and let [22Xg ∈ M[122X. Since [22XH[122X is
  the disjoint union of [22XV[122X, [22XH' ∖ V[122X, and three [22XH[122X-conjugates of [22XU ∖ V[122X, we get[133X
  
    1_H^M(g) = [M:H] ⋅ |g^M ∩ H| / |g^M|
             = [M:H] ⋅ ( |g^M ∩ V| + 3 |g^M ∩ U \ V|
                                   + |g^M ∩ H' \ V| ) / |g^M|
             = [M:H] ⋅ ( 3 |g^M ∩ U| - 2 |g^M ∩ V|
                                     + |g^M ∩ H' \ V| ) / |g^M|
             = 1_U^M(g) - 1/3 ⋅ 1_V^G(g)
                        + [M:H] ⋅ |g^M ∩ H' \ V| / |g^M| .
  
  [33X[0;0YPossible  constituents of [22X1_H^M[122X are those rational irreducible characters of
  [22XM[122X that are constituents of [22Xπ^M[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xconstit:= Filtered( RationalizedMat( Irr( m ) ),[127X[104X
    [4X[25X>[125X [27X                       chi -> ScalarProduct( m, chi, pi ) <> 0 );;[127X[104X
  [4X[32X[104X
  
  [33X[0;0YNow  we compute the possible permutation characters that have the prescribed
  values, are compatible with the given lower bounds for values, and have only
  constituents in the given list.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xcand:= PermChars( m,[127X[104X
    [4X[25X>[125X [27X     rec( torso:= cand, chars:= constit,[127X[104X
    [4X[25X>[125X [27X          lower:= ShallowCopy( pi - 1/3 * phi ),[127X[104X
    [4X[25X>[125X [27X          normalsubgroup:= [ 1 .. NrConjugacyClasses( m ) ],[127X[104X
    [4X[25X>[125X [27X          nonfaithful:= TrivialCharacter( m ) ) );[127X[104X
    [4X[28X[ Character( CharacterTable( "M" ),[128X[104X
    [4X[28X  [ 16009115629875684006343550944921875, 7774182899642733721875, [128X[104X
    [4X[28X      120168544413337875, 4436049512692980, 215448838605, [128X[104X
    [4X[28X      131873639625, 760550656275, 110042727795, 943894035, 568854195, [128X[104X
    [4X[28X      1851609375, 0, 4680311220, 405405, 78624756, 14467005, 178605, [128X[104X
    [4X[28X      248265, 874650, 0, 76995, 591163, 224055, 34955, 29539, 20727, [128X[104X
    [4X[28X      0, 0, 375375, 15775, 0, 0, 0, 495, 116532, 3645, 62316, 1017, [128X[104X
    [4X[28X      11268, 357, 1701, 45, 117, 705, 0, 0, 4410, 1498, 0, 3780, 810, [128X[104X
    [4X[28X      0, 0, 83, 135, 31, 0, 0, 0, 0, 0, 0, 0, 255, 195, 0, 215, 0, 0, [128X[104X
    [4X[28X      210, 0, 42, 0, 35, 15, 1, 1, 160, 48, 9, 92, 25, 9, 9, 5, 1, [128X[104X
    [4X[28X      21, 0, 0, 0, 0, 0, 98, 74, 42, 0, 0, 0, 120, 76, 10, 0, 0, 0, [128X[104X
    [4X[28X      0, 0, 1, 1, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 5, 3, 0, [128X[104X
    [4X[28X      0, 0, 18, 0, 10, 0, 3, 3, 0, 1, 1, 1, 1, 0, 0, 2, 0, 0, 0, 0, [128X[104X
    [4X[28X      0, 0, 2, 0, 0, 0, 0, 0, 6, 12, 0, 0, 2, 0, 0, 0, 2, 0, 0, 1, 1, [128X[104X
    [4X[28X      0, 0, 0, 0, 0, 0, 0, 2, 0, 2, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, [128X[104X
    [4X[28X      0, 0, 0, 0, 0, 0, 0 ] ) ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YThere is only one candidate, so we have found the permutation character.[133X
  
  
  [1X8.16-2 [33X[0;0YThe Subgroup [22X2^3.2^6.2^12.2^18.(L_3(2) × 3.S_6)[122X[101X[1X (September 2009)[133X[101X
  
  [33X[0;0YAccording  to the Atlas of Finite Groups [CCN+85, p. 234], the Monster group
  [22XM[122X has a class of maximal subgroups [22XH[122X of the type [22X2^3.2^6.2^12.2^18.(L_3(2) ×
  3.S_6)[122X.  Currently  the character table of [22XH[122X and the class fusion into [22XM[122X are
  not  available in [5XGAP[105X. We are interested in the permutation character [22X1_H^G[122X,
  and we will compute it without this information.[133X
  
  [33X[0;0YThe subgroup [22XH[122X normalizes an elementary abelian group [22XE[122X of order eight whose
  involutions  lie  in  the class [10X2B[110X. We fix an involution [22Xz[122X in [22XE[122X, and set [22XG =
  C_M(z)[122X,  [22XU  =  C_H(z)[122X,  and  [22XV  = C_H(E)[122X. Further, let [22XN[122X be the extraspecial
  normal subgroup of order [22X2^25[122X in [22XG[122X.[133X
  
  [33X[0;0YSo  [22XG[122X  has  the  structure  [22X2^{1+24}_+.Co_1[122X, and [22XU[122X has index seven in [22XH[122X. The
  order of [22XN U / N[122X is a multiple of [22X2^{3+6+12+18-25} ⋅ |L_3(2)| ⋅ |3.S_6| / 7[122X,
  and [22XN U / N[122X occurs as a subgroup of [22XG / N ≅ Co_1[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xco1:= CharacterTable( "Co1" );;[127X[104X
    [4X[25Xgap>[125X [27Xorder:= 2^(3+6+12+18-25) * 168 * 3 * Factorial( 6 ) / 7;[127X[104X
    [4X[28X849346560[128X[104X
    [4X[25Xgap>[125X [27Xmaxes:= List( Maxes( co1 ), CharacterTable );;[127X[104X
    [4X[25Xgap>[125X [27Xfilt:= Filtered( maxes, t -> Size( t ) mod order = 0 );[127X[104X
    [4X[28X[ CharacterTable( "2^(1+8)+.O8+(2)" ), [128X[104X
    [4X[28X  CharacterTable( "2^(4+12).(S3x3S6)" ) ][128X[104X
    [4X[25Xgap>[125X [27XList( filt, t -> Size( t ) / order );[127X[104X
    [4X[28X[ 105, 1 ][128X[104X
    [4X[25Xgap>[125X [27Xo8p2:= CharacterTable( "O8+(2)" );;[127X[104X
    [4X[25Xgap>[125X [27XPermChars( o8p2, rec( torso:= [ 105 ] ) );[127X[104X
    [4X[28X[  ][128X[104X
    [4X[25Xgap>[125X [27Xk:= filt[2];;[127X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  list  of maximal subgroups of [22XCo_1[122X (see [CCN+85, p. 183]) tells us that
  [22XNU / N[122X is a maximal subgroup [22XK[122X of [22XCo_1[122X and has the structure [22X2^{4+12}.(S_3 ×
  3.S_6)[122X.  (Note that the group [22XO_8^+(2)[122X has no proper subgroup of index [22X105[122X.)
  In particular, [22XU[122X contains [22XN[122X and thus [22XU/N ≅ K[122X.[133X
  
  [33X[0;0YLet  [22XC[122X  be  the set of elements in [22XM[122X whose order is not divisible by [22X7[122X. Then
  [22X(1_H)_{C  ∩  H} = (1_U^H - 1/3 1_VN^H + 1/21 1_V^H)_{C ∩ H}[122X holds, as we can
  see from computations with [22XH/V ≅ L_3(2)[122X, as follows.[133X
  
  [33X[0;0YSo S4, V4, 1 suffice! -->[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xf:= CharacterTable( "L3(2)" );[127X[104X
    [4X[28XCharacterTable( "L3(2)" )[128X[104X
    [4X[25Xgap>[125X [27XOrdersClassRepresentatives( f );[127X[104X
    [4X[28X[ 1, 2, 3, 4, 7, 7 ][128X[104X
    [4X[25Xgap>[125X [27Xdeg7:= PermChars( f, 7 );[127X[104X
    [4X[28X[ Character( CharacterTable( "L3(2)" ), [ 7, 3, 1, 1, 0, 0 ] ) ][128X[104X
    [4X[25Xgap>[125X [27Xdeg42:= PermChars( f, 42 );[127X[104X
    [4X[28X[ Character( CharacterTable( "L3(2)" ), [ 42, 2, 0, 2, 0, 0 ] ), [128X[104X
    [4X[28X  Character( CharacterTable( "L3(2)" ), [ 42, 6, 0, 0, 0, 0 ] ) ][128X[104X
    [4X[25Xgap>[125X [27Xdeg168:= PermChars( f, 168 );[127X[104X
    [4X[28X[ Character( CharacterTable( "L3(2)" ), [ 168, 0, 0, 0, 0, 0 ] ) ][128X[104X
    [4X[25Xgap>[125X [27Xdeg7[1] - 1/3 * deg42[2] + 1/21 * deg168[1];[127X[104X
    [4X[28XClassFunction( CharacterTable( "L3(2)" ), [ 1, 1, 1, 1, 0, 0 ] )[128X[104X
  [4X[32X[104X
  
  [33X[0;0Y(Note  that  [22XVN/V[122X  is  a  Klein four group, and there is only one transitive
  permutation character of [22XL_3(2)[122X that is induced from such subgroups.)[133X
  
  [33X[0;0YThe  character  table  of  [22XG[122X  is  available  in  [5XGAP[105X,  so we can compute the
  permutation  character  [22Xπ  =  1_U^G[122X  by  computing the primitive permutation
  character  [22X1_K^{Co_1}[122X, identifying it with [22X1_{U/N}^{G/N}[122X, and then inflating
  this character to [22XG[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xm:= CharacterTable( "M" );[127X[104X
    [4X[28XCharacterTable( "M" )[128X[104X
    [4X[25Xgap>[125X [27Xg:= CharacterTable( "MC2B" );[127X[104X
    [4X[28XCharacterTable( "2^1+24.Co1" )[128X[104X
    [4X[25Xgap>[125X [27Xpi:= RestrictedClassFunction( TrivialCharacter( k )^co1, g );;[127X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  permutation  character  [22Xψ  = 1_VN^G[122X can be computed as the inflation of
  [22X1_{VN/N}^{G/N} = (1_{VN/N}^{U/N})^{G/N}[122X, where [22X1_{VN/N}^{U/N}[122X is a character
  of [22XK[122X that can be identified with the regular permutation character of [22XU/VN ≅
  S_3[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xnsg:= ClassPositionsOfNormalSubgroups( k );;[127X[104X
    [4X[25Xgap>[125X [27Xnsgsizes:= List( nsg, x -> Sum( SizesConjugacyClasses( k ){ x } ) );;[127X[104X
    [4X[25Xgap>[125X [27Xnn:= nsg[ Position( nsgsizes, Size( k ) / 6 ) ];;[127X[104X
    [4X[25Xgap>[125X [27Xpsi:= 0 * [ 1 .. NrConjugacyClasses( k ) ];;[127X[104X
    [4X[25Xgap>[125X [27Xfor i in nn do[127X[104X
    [4X[25X>[125X [27X     psi[i]:= 6;[127X[104X
    [4X[25X>[125X [27X   od;[127X[104X
    [4X[25Xgap>[125X [27Xpsi:= InducedClassFunction( k, psi, co1 );;[127X[104X
    [4X[25Xgap>[125X [27Xpsi:= RestrictedClassFunction( psi, g );;[127X[104X
  [4X[32X[104X
  
  [33X[0;0YNext  we  consider the permutation character [22Xϕ = 1_V^G[122X. The group [22XV[122X does not
  contain  [22XN[122X  because  [22XK[122X  does  not have a factor group of the type [22XS_4[122X. But [22XV[122X
  contains [22XZ[122X because [22XU/V[122X is centerless. So we can regard [22Xϕ[122X as the inflation of
  [22X1_{V/Z}^{G/Z}[122X from [22XG/Z[122X to [22XG[122X, i. e., we can perform the computations with the
  character table of the factor group [22XG/Z[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xzclasses:= ClassPositionsOfCentre( g );;[127X[104X
    [4X[25Xgap>[125X [27Xgmodz:= g / zclasses;[127X[104X
    [4X[28XCharacterTable( "2^1+24.Co1/[ 1, 2 ]" )[128X[104X
    [4X[25Xgap>[125X [27Xinvmap:= InverseMap( GetFusionMap( g, gmodz ) );;[127X[104X
    [4X[25Xgap>[125X [27Xpsibar:= CompositionMaps( psi, invmap );;[127X[104X
  [4X[32X[104X
  
  [33X[0;0YSince  [22Xϕ(g) = [G:V] ⋅ |g^G ∩ V| / |g^G|[122X holds for [22Xg ∈ G[122X, and since [22Xg^G ∩ V ⊆
  g^G  ∩ VN[122X, with equality if [22Xg[122X has odd order, we get [22Xϕ(g) = 4 ⋅ ψ(g)[122X if [22Xg[122X has
  odd order, and [22Xϕ(g) = 0[122X if [22Xψ(g) = 0[122X.[133X
  
  [33X[0;0YWe  want  to  compute the possible permutation characters with these values.
  This is easier if we [21Xgo down[121X from [22XVN[122X to [22XV[122X in two steps.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xfactorders:= OrdersClassRepresentatives( gmodz );;[127X[104X
    [4X[25Xgap>[125X [27Xphibar:= [];;[127X[104X
    [4X[25Xgap>[125X [27Xupperphibar:= [];;[127X[104X
    [4X[25Xgap>[125X [27Xfor i in [ 1 .. NrConjugacyClasses( gmodz ) ] do[127X[104X
    [4X[25X>[125X [27X     if factorders[i] mod 2 = 1 then[127X[104X
    [4X[25X>[125X [27X       phibar[i]:= 2 * psibar[i];[127X[104X
    [4X[25X>[125X [27X     elif psibar[i] = 0 then[127X[104X
    [4X[25X>[125X [27X       phibar[i]:= 0;[127X[104X
    [4X[25X>[125X [27X     fi;[127X[104X
    [4X[25X>[125X [27X     upperphibar[i]:= 2 * psibar[i];[127X[104X
    [4X[25X>[125X [27X   od;[127X[104X
    [4X[25Xgap>[125X [27Xcand:= PermChars( gmodz, rec( torso:= phibar,[127X[104X
    [4X[25X>[125X [27X            upper:= upperphibar,[127X[104X
    [4X[25X>[125X [27X            normalsubgroup:= [ 1 .. NrConjugacyClasses( gmodz ) ],[127X[104X
    [4X[25X>[125X [27X            nonfaithful:= TrivialCharacter( gmodz ) ) );;[127X[104X
    [4X[25Xgap>[125X [27XLength( cand );[127X[104X
    [4X[28X3[128X[104X
  [4X[32X[104X
  
  [33X[0;0YOne  of  the  candidates computed in this first step is excluded by the fact
  that it is induced from a subgroup that contains [22XN/Z[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xnn:= First( ClassPositionsOfNormalSubgroups( gmodz ),[127X[104X
    [4X[25X>[125X [27X               x -> Sum( SizesConjugacyClasses( gmodz ){x} ) = 2^24 );[127X[104X
    [4X[28X[ 1 .. 4 ][128X[104X
    [4X[25Xgap>[125X [27Xcont:= PermCharInfo( gmodz, cand ).contained;;[127X[104X
    [4X[25Xgap>[125X [27Xcand:= cand{ Filtered( [ 1 .. Length( cand ) ],[127X[104X
    [4X[25X>[125X [27X                          i -> Sum( cont[i]{ nn } ) < 2^24 ) };;[127X[104X
    [4X[25Xgap>[125X [27XLength( cand );[127X[104X
    [4X[28X2[128X[104X
  [4X[32X[104X
  
  [33X[0;0YNow  we  run  the second step. After excluding the candidates that cannot be
  induced from subgroups whose intersection with [22XN/Z[122X has index four in [22XN/Z[122X, we
  get four solutions.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xposs:= [];;[127X[104X
    [4X[25Xgap>[125X [27Xfor v in cand do[127X[104X
    [4X[25X>[125X [27X     phibar:= [];[127X[104X
    [4X[25X>[125X [27X     upperphibar:= [];[127X[104X
    [4X[25X>[125X [27X     for i in [ 1 .. NrConjugacyClasses( gmodz ) ] do[127X[104X
    [4X[25X>[125X [27X       if factorders[i] mod 2 = 1 then[127X[104X
    [4X[25X>[125X [27X         phibar[i]:= 2 * v[i];[127X[104X
    [4X[25X>[125X [27X       elif v[i] = 0 then[127X[104X
    [4X[25X>[125X [27X         phibar[i]:= 0;[127X[104X
    [4X[25X>[125X [27X       fi;[127X[104X
    [4X[25X>[125X [27X       upperphibar[i]:= 2 * v[i];[127X[104X
    [4X[25X>[125X [27X     od;[127X[104X
    [4X[25X>[125X [27X     Append( poss, PermChars( gmodz, rec( torso:= phibar,[127X[104X
    [4X[25X>[125X [27X                     upper:= upperphibar,[127X[104X
    [4X[25X>[125X [27X                     normalsubgroup:= [ 1 .. NrConjugacyClasses( gmodz ) ],[127X[104X
    [4X[25X>[125X [27X                     nonfaithful:= TrivialCharacter( gmodz ) ) ) );[127X[104X
    [4X[25X>[125X [27X   od;[127X[104X
    [4X[25Xgap>[125X [27XLength( poss );[127X[104X
    [4X[28X6[128X[104X
    [4X[25Xgap>[125X [27Xcont:= PermCharInfo( gmodz, poss ).contained;;[127X[104X
    [4X[25Xgap>[125X [27Xposs:= poss{ Filtered( [ 1 .. Length( poss ) ],[127X[104X
    [4X[25X>[125X [27X                          i -> Sum( cont[i]{ nn } ) < 2^23 ) };;[127X[104X
    [4X[25Xgap>[125X [27XLength( poss );[127X[104X
    [4X[28X4[128X[104X
    [4X[25Xgap>[125X [27Xphicand:= RestrictedClassFunctions( poss, g );;[127X[104X
  [4X[32X[104X
  
  [33X[0;0YSince  we have several candidates for [22X1_V^G[122X, we form the linear combinations
  for all these candidates.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xphicand:= RestrictedClassFunctions( poss, g );;[127X[104X
    [4X[25Xgap>[125X [27Xphicand:= InducedClassFunctions( phicand, m );;[127X[104X
    [4X[25Xgap>[125X [27Xpsi:= psi^m;;[127X[104X
    [4X[25Xgap>[125X [27Xpi:= pi^m;;[127X[104X
    [4X[25Xgap>[125X [27Xcand:= List( phicand,[127X[104X
    [4X[25X>[125X [27X            phi -> ShallowCopy( pi - 1/3 * psi + 1/21 * phi ) );;[127X[104X
    [4X[25Xgap>[125X [27Xmorders:= OrdersClassRepresentatives( m );;[127X[104X
    [4X[25Xgap>[125X [27Xfor x in cand do[127X[104X
    [4X[25X>[125X [27X     for i in [ 1 .. Length( morders ) ] do[127X[104X
    [4X[25X>[125X [27X       if morders[i] mod 7 = 0 then[127X[104X
    [4X[25X>[125X [27X         Unbind( x[i] );[127X[104X
    [4X[25X>[125X [27X       fi;[127X[104X
    [4X[25X>[125X [27X     od;[127X[104X
    [4X[25X>[125X [27X   od;[127X[104X
  [4X[32X[104X
  
  [33X[0;0YExactly one of the candidates has only integral values.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xcand:= Filtered( cand, x -> ForAll( x, IsInt ) );[127X[104X
    [4X[28X[ [ 4050306254358548053604918389065234375, 148844831270071996434375, [128X[104X
    [4X[28X      2815847622206994375, 14567365753025085, 3447181417680, [128X[104X
    [4X[28X      659368198125, 3520153823175, 548464353255, 5706077895, [128X[104X
    [4X[28X      3056566695, 264515625, 0, 19572895485, 6486480, 186109245, [128X[104X
    [4X[28X      61410960, 758160, 688365,,, 172503, 1264351, 376155, 137935, [128X[104X
    [4X[28X      99127, 52731, 0, 0, 119625, 3625, 0, 0, 0, 0, 402813, 29160, [128X[104X
    [4X[28X      185301, 2781, 21069, 1932, 4212, 360, 576, 1125, 0, 0,,,, 2160, [128X[104X
    [4X[28X      810, 0, 0, 111, 179, 43, 0, 0, 0, 0, 0, 0, 0, 185, 105, 0, 65, [128X[104X
    [4X[28X      0, 0,,,,, 0, 0, 0, 0, 337, 105, 36, 157, 37, 18, 18, 16, 4, 21, [128X[104X
    [4X[28X      0, 0, 0, 0, 0,,,,, 0, 0, 60, 40, 10, 0, 0, 0, 0, 0, 1, 1, 0, 0, [128X[104X
    [4X[28X      0,,, 0, 0, 0, 0, 0, 0, 0, 0, 0, 5, 1, 0, 0, 0,,,,, 0, 0, 0, 0, [128X[104X
    [4X[28X      0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0,,,, 0, 0, 0, 6, 8, 0, 0, 2, [128X[104X
    [4X[28X      0, 0, 0, 0, 0, 0, 0, 0,,, 0, 0, 0, 0, 0,,,, 0, 0, 0, 0, 0, 0, [128X[104X
    [4X[28X      0, 0, 0, 0, 0, 0, 0, 0,, 0 ] ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YPossible  constituents of [22X1_H^M[122X are those rational irreducible characters of
  [22XM[122X that are constituents of [22Xπ^M[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xconstit:= Filtered( RationalizedMat( Irr( m ) ),[127X[104X
    [4X[25X>[125X [27X                       chi -> ScalarProduct( m, chi, pi ) <> 0 );;[127X[104X
  [4X[32X[104X
  
  [33X[0;0YNow  we compute the possible permutation characters that have the prescribed
  values and have only constituents in the given list.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xcand:= PermChars( m, rec( torso:= cand[1], chars:= constit ) );[127X[104X
    [4X[28X[ Character( CharacterTable( "M" ),[128X[104X
    [4X[28X  [ 4050306254358548053604918389065234375, 148844831270071996434375, [128X[104X
    [4X[28X      2815847622206994375, 14567365753025085, 3447181417680, [128X[104X
    [4X[28X      659368198125, 3520153823175, 548464353255, 5706077895, [128X[104X
    [4X[28X      3056566695, 264515625, 0, 19572895485, 6486480, 186109245, [128X[104X
    [4X[28X      61410960, 758160, 688365, 58310, 0, 172503, 1264351, 376155, [128X[104X
    [4X[28X      137935, 99127, 52731, 0, 0, 119625, 3625, 0, 0, 0, 0, 402813, [128X[104X
    [4X[28X      29160, 185301, 2781, 21069, 1932, 4212, 360, 576, 1125, 0, 0, [128X[104X
    [4X[28X      1302, 294, 0, 2160, 810, 0, 0, 111, 179, 43, 0, 0, 0, 0, 0, 0, [128X[104X
    [4X[28X      0, 185, 105, 0, 65, 0, 0, 224, 0, 14, 0, 0, 0, 0, 0, 337, 105, [128X[104X
    [4X[28X      36, 157, 37, 18, 18, 16, 4, 21, 0, 0, 0, 0, 0, 70, 38, 14, 0, [128X[104X
    [4X[28X      0, 0, 60, 40, 10, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 10, 0, 0, 0, 0, [128X[104X
    [4X[28X      0, 0, 0, 0, 0, 0, 5, 1, 0, 0, 0, 24, 0, 6, 0, 0, 0, 0, 0, 0, 0, [128X[104X
    [4X[28X      0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 6, 8, 0, 0, 2, [128X[104X
    [4X[28X      0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 4, 0, 2, 0, 0, 0, [128X[104X
    [4X[28X      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0 ] ) ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YThere is only one candidate, so we have found the permutation character.[133X
  
  
  [1X8.16-3 [33X[0;0YThe Subgroup [22X2^5.2^10.2^20.(S_3 × L_5(2))[122X[101X[1X (October 2009)[133X[101X
  
  [33X[0;0YAccording  to the Atlas of Finite Groups [CCN+85, p. 234], the Monster group
  [22XM[122X  has  a  class  of  maximal  subgroups  [22XH[122X of the type [22X2^5.2^10.2^20.(S_3 ×
  L_5(2))[122X.  Currently the character table of [22XH[122X and the class fusion into [22XM[122X are
  not  available in [5XGAP[105X. We are interested in the permutation character [22X1_H^G[122X,
  and we will compute it without this information.[133X
  
  [33X[0;0YThe  subgroup  [22XH[122X  normalizes an elementary abelian group [22XE[122X of order [22X32[122X whose
  involutions  lie  in  the class [10X2B[110X. We fix an involution [22Xz[122X in [22XE[122X, and set [22XG =
  C_M(z)[122X,  [22XU  =  C_H(z)[122X,  and  [22XV  = C_H(E)[122X. Further, let [22XN[122X be the extraspecial
  normal subgroup of order [22X2^25[122X in [22XG[122X.[133X
  
  [33X[0;0YSo  [22XG[122X  has the structure [22X2^{1+24}_+.Co_1[122X, and [22XU[122X has index [22X31[122X in [22XH[122X. The order
  of  [22XN U / N[122X is a multiple of [22X2^{5+10+20-25} ⋅ |L_5(2)| ⋅ |S_3| / 31[122X, and [22XN U
  / N[122X occurs as a subgroup of [22XG / N ≅ Co_1[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xco1:= CharacterTable( "Co1" );;[127X[104X
    [4X[25Xgap>[125X [27Xorder:= 2^35*Size( CharacterTable( "L5(2)" ) )*6 / 2^25 / 31;[127X[104X
    [4X[28X1981808640[128X[104X
    [4X[25Xgap>[125X [27Xmaxes:= List( Maxes( co1 ), CharacterTable );;[127X[104X
    [4X[25Xgap>[125X [27Xfilt:= Filtered( maxes, t -> Size( t ) mod order = 0 );[127X[104X
    [4X[28X[ CharacterTable( "2^11:M24" ), CharacterTable( "2^(1+8)+.O8+(2)" ), [128X[104X
    [4X[28X  CharacterTable( "2^(2+12):(A8xS3)" ) ][128X[104X
    [4X[25Xgap>[125X [27XList( filt, t -> Size( t ) / order );[127X[104X
    [4X[28X[ 253, 45, 1 ][128X[104X
    [4X[25Xgap>[125X [27Xm24:= CharacterTable( "M24" );;[127X[104X
    [4X[25Xgap>[125X [27Xcand:= PermChars( m24, rec( torso:=[ 253 ] ) );[127X[104X
    [4X[28X[ Character( CharacterTable( "M24" ),[128X[104X
    [4X[28X  [ 253, 29, 13, 10, 1, 5, 5, 1, 3, 2, 1, 1, 1, 1, 3, 0, 2, 1, 1, 1, [128X[104X
    [4X[28X      0, 0, 1, 1, 0, 0 ] ) ][128X[104X
    [4X[25Xgap>[125X [27XTestPerm5( m24, cand, m24 mod 11 );[127X[104X
    [4X[28X[  ][128X[104X
    [4X[25Xgap>[125X [27XPermChars( CharacterTable( "O8+(2)" ), rec( torso:=[ 45 ] ) );[127X[104X
    [4X[28X[  ][128X[104X
    [4X[25Xgap>[125X [27Xk:= filt[3];;[127X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  list  of maximal subgroups of [22XCo_1[122X (see [CCN+85, p. 183]) tells us that
  [22XNU / N[122X is a maximal subgroup [22XK[122X of [22XCo_1[122X and has the structure [22X2^{2+12}.(A_8 ×
  S_3)[122X.  (Note  that the group [22XM_24[122X has no proper subgroup of index [22X253[122X, which
  is  shown  above using the [22X11[122X-modular Brauer table of [22XM_24[122X. Furthermore, the
  group [22XO_8^+(2)[122X has no subgroup of index [22X45[122X.) In particular, [22XU[122X contains [22XN[122X and
  thus [22XU/N ≅ K[122X.[133X
  
  [33X[0;0YLet  [22XC[122X be the set of elements in [22XM[122X whose order is not divisible by [22X31[122X or [22X21[122X.
  We  want  to  find  an  index  set  [22XI[122X and subgroups [22XX_i[122X, for [22Xi ∈ I[122X, with the
  property that [22XV ≤ X_i ≤ U[122X and[133X
  
  
  [24X[33X[0;6Y(1_H)_{C ∩ H} = ( ∑_{i ∈ I} c_i 1_{X_i}^H )_{C ∩ H}[133X[124X
  
  [33X[0;0Yholds  for suitable rational integers [22Xc_i[122X. Let [22XW[122X be the full preimage of the
  elementary  normal  subgroup  of order [22X16[122X in [22XU/V ≅ 2^4.A_8[122X under the natural
  epimorphism  from  [22XU[122X  to [22XU/V[122X, and set [22XI_1 = { i ∈ I; W ≤ X_i }[122X and [22XI_2 = I ∖
  I_1[122X.[133X
  
  [33X[0;0YUsing  the  known  table  of marks of [22XU/V[122X, we will find a solution such that
  [22X[W:(W  ∩  X_i)]  =  2[122X  for  all  [22Xi  ∈  I_2[122X. First we compute the permutation
  characters [22X1_S^{U/V}[122X for all subgroups [22XS[122X of [22XU/V[122X that contain [22XW/V[122X, and induce
  them to [22XH/V[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xsubtbl:= CharacterTable( "2^4:A8" );;[127X[104X
    [4X[25Xgap>[125X [27Xsubtom:= TableOfMarks( subtbl );;[127X[104X
    [4X[25Xgap>[125X [27Xperms:= PermCharsTom( subtbl, subtom );;[127X[104X
    [4X[25Xgap>[125X [27Xnsg:= ClassPositionsOfNormalSubgroups( subtbl );[127X[104X
    [4X[28X[ [ 1 ], [ 1, 2 ], [ 1 .. 25 ] ][128X[104X
    [4X[25Xgap>[125X [27Xabove:= Filtered( perms, x -> x[1] = x[2] );;[127X[104X
    [4X[25Xgap>[125X [27Xtbl:= CharacterTable( "L5(2)" );;[127X[104X
    [4X[25Xgap>[125X [27Xabove:= Set( Induced( subtbl, tbl, above ) );;[127X[104X
  [4X[32X[104X
  
  [33X[0;0YNext  we compute the permutation characters [22X1_S^{U/V}[122X for all subgroups [22XS[122X of
  [22XU/V[122X  whose intersection with [22XW/V[122X has index two in [22XW/V[122X. Afterwards we exclude
  certain  subgroups  that would slow down later computations, and induce also
  these characters to [22XH/V[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xindex2:= Filtered( perms,[127X[104X
    [4X[25X>[125X [27X     x -> Sum( PermCharInfo( subtbl, [x] ).contained[1]{ [1,2] } ) = 8 );;[127X[104X
    [4X[25Xgap>[125X [27Xindex2:= Filtered( index2, x -> not x[1] in [ 630, 840, 1260, 1680 ] );;[127X[104X
    [4X[25Xgap>[125X [27Xindex2:= Set( Induced( subtbl, tbl, index2 ) );;[127X[104X
  [4X[32X[104X
  
  [33X[0;0YNow   we   induce  the  permutation  characters  to  [22XH/V[122X,  and  compute  the
  coefficients of a linear combination as desired.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xorders:= OrdersClassRepresentatives( tbl );;[127X[104X
    [4X[25Xgap>[125X [27Xgoodclasses:= Filtered( [ 1 .. NrConjugacyClasses( tbl ) ],[127X[104X
    [4X[25X>[125X [27X                           i -> not orders[i] in [ 21, 31 ] );[127X[104X
    [4X[28X[ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19 ][128X[104X
    [4X[25Xgap>[125X [27Xmatrix:= List( Concatenation( above, index2 ), x -> x{ goodclasses } );;[127X[104X
    [4X[25Xgap>[125X [27Xsol:= SolutionMat( matrix,[127X[104X
    [4X[25X>[125X [27X             ListWithIdenticalEntries( Length( goodclasses ), 1 ) );[127X[104X
    [4X[28X[ 692/651, 57/217, -78/217, -26/217, 0, 74/651, 11/217, 0, 3/217, [128X[104X
    [4X[28X  151/651, 0, 22/651, 0, 0, 0, -11/217, 0, 0, 0, 0, 0, 0, 0, 0, [128X[104X
    [4X[28X  -115/651, 0, -3/31, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, [128X[104X
    [4X[28X  0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, [128X[104X
    [4X[28X  0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, [128X[104X
    [4X[28X  0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -34/93, [128X[104X
    [4X[28X  -11/651, 0, 2/21, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1/31, 0, 0, 0, 0, [128X[104X
    [4X[28X  0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, [128X[104X
    [4X[28X  0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, [128X[104X
    [4X[28X  0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, [128X[104X
    [4X[28X  0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, [128X[104X
    [4X[28X  0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, [128X[104X
    [4X[28X  0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ][128X[104X
    [4X[25Xgap>[125X [27Xnonzero:= Filtered( [ 1 .. Length( sol ) ], i -> sol[i] <> 0 );[127X[104X
    [4X[28X[ 1, 2, 3, 4, 6, 7, 9, 10, 12, 16, 25, 27, 106, 107, 109, 120 ][128X[104X
    [4X[25Xgap>[125X [27Xsol:= sol{ nonzero };;[127X[104X
  [4X[32X[104X
  
  [33X[0;0YNow  we  transfer  this linear combination to the character tables which are
  given in our situation.[133X
  
  [33X[0;0YThose  constituents  that  are  induced  from  subgroups of [22XH[122X above [22XW[122X can be
  identified  uniquely  via  their  degrees  and their values distribution; we
  compute  these characters in the character table of [22XU/W[122X obtained as a factor
  table  of  the character table of [22XU/N[122X, lift them back to [22XU/N[122X, induce them to
  [22XG/N[122X, inflate them to [22XG[122X, and then induce them fo [22XM[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xa8degrees:= List( above{ Filtered( nonzero,[127X[104X
    [4X[25X>[125X [27X                                x -> x <= Length( above ) ) },[127X[104X
    [4X[25X>[125X [27X                     x -> x[1] ) / 31;[127X[104X
    [4X[28X[ 1, 8, 15, 28, 56, 56, 70, 105, 120, 168, 336, 336 ][128X[104X
    [4X[25Xgap>[125X [27Xa8tbl:= subtbl / [ 1, 2 ];;[127X[104X
    [4X[25Xgap>[125X [27Xinvtoa8:= InverseMap( GetFusionMap( subtbl, a8tbl ) );;[127X[104X
    [4X[25Xgap>[125X [27Xnsg:= ClassPositionsOfNormalSubgroups( k );;[127X[104X
    [4X[25Xgap>[125X [27Xnn:= First( nsg, x -> Sum( SizesConjugacyClasses( k ){ x } ) = 6*2^14 );;[127X[104X
    [4X[25Xgap>[125X [27Xa8tbl_other:= k / nn;;[127X[104X
    [4X[25Xgap>[125X [27Xg:= CharacterTable( "MC2B" );[127X[104X
    [4X[28XCharacterTable( "2^1+24.Co1" )[128X[104X
    [4X[25Xgap>[125X [27Xconstit:= [];;[127X[104X
    [4X[25Xgap>[125X [27Xfor i in [ 1 .. Length( a8degrees ) ] do[127X[104X
    [4X[25X>[125X [27X     cand:= PermChars( a8tbl_other, rec( torso:= [ a8degrees[i] ] ) );[127X[104X
    [4X[25X>[125X [27X     filt:= Filtered( perms, x -> x^tbl = above[ nonzero[i] ] );[127X[104X
    [4X[25X>[125X [27X     filt:= List( filt, x -> CompositionMaps( x, invtoa8 ) );[127X[104X
    [4X[25X>[125X [27X     cand:= Filtered( cand,[127X[104X
    [4X[25X>[125X [27X              x -> ForAny( filt, y -> Collected( x ) = Collected(y) ) );[127X[104X
    [4X[25X>[125X [27X     Add( constit, List( Induced( Restricted( Induced([127X[104X
    [4X[25X>[125X [27X       Restricted( cand, k ), co1 ), g ), m ), ValuesOfClassFunction ) );[127X[104X
    [4X[25X>[125X [27X   od;[127X[104X
    [4X[25Xgap>[125X [27XList( constit, Length );[127X[104X
    [4X[28X[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YDealing  with the remaining constituents is more involved. For a permutation
  character [22X1_{X/V}^{U/V}[122X, we compute [22X1_{WX/V}^{U/V}[122X, a character whose degree
  is  half  as  large  and  which  can be regarded as a character of [22XU/W[122X. This
  character  can  be treated like the ones above: We lift it to [22XU/N[122X, induce it
  to  [22XG/N[122X,  and  inflate it to [22XG/Z(G)[122X; let this character be [22X1_Y^{G/Z(G)}[122X, for
  some  subgroup  [22XY[122X.  Then  we  compute the possible permutation characters of
  [22XG/Z(G)[122X  that  can  be induced from a subgroup of index two inside [22XY[122X, inflate
  these characters to [22XG[122X and then induce them to [22XM[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xdowndegrees:= List( index2{ Filtered( nonzero,[127X[104X
    [4X[25X>[125X [27X                                   x -> x > Length( above ) )[127X[104X
    [4X[25X>[125X [27X                               - Length( above ) },[127X[104X
    [4X[25X>[125X [27X                       x -> x[1] ) / 31;[127X[104X
    [4X[28X[ 30, 210, 210, 1920 ][128X[104X
    [4X[25Xgap>[125X [27Xf:= g / ClassPositionsOfCentre( g );;[127X[104X
    [4X[25Xgap>[125X [27Xforders:= OrdersClassRepresentatives( f );;[127X[104X
    [4X[25Xgap>[125X [27Xinv:= InverseMap( GetFusionMap( g, f ) );;[127X[104X
    [4X[25Xgap>[125X [27Xfor j in [ 1 .. Length( downdegrees ) ] do[127X[104X
    [4X[25X>[125X [27X     chars:= [];[127X[104X
    [4X[25X>[125X [27X     cand:= PermChars( a8tbl_other, rec( torso:= [ downdegrees[j]/2 ] ) );[127X[104X
    [4X[25X>[125X [27X     filt:= Filtered( perms, x -> x^tbl = index2[ nonzero[[127X[104X
    [4X[25X>[125X [27X                  j + Length( a8degrees ) ] - Length( above ) ] );[127X[104X
    [4X[25X>[125X [27X     filt:= Induced( subtbl, a8tbl, filt,[127X[104X
    [4X[25X>[125X [27X                     GetFusionMap( subtbl, a8tbl ));[127X[104X
    [4X[25X>[125X [27X     cand:= Filtered( cand, x -> ForAny( filt,[127X[104X
    [4X[25X>[125X [27X                y -> Collected( x ) = Collected( y ) ) );[127X[104X
    [4X[25X>[125X [27X     cand:= Restricted( Induced( Restricted( cand, k ), co1 ), g );[127X[104X
    [4X[25X>[125X [27X     for chi in cand do[127X[104X
    [4X[25X>[125X [27X       cchi:= CompositionMaps( chi, inv );[127X[104X
    [4X[25X>[125X [27X       upper:= [];[127X[104X
    [4X[25X>[125X [27X       pphi:= [];[127X[104X
    [4X[25X>[125X [27X       for i in [ 1 .. NrConjugacyClasses( f ) ] do[127X[104X
    [4X[25X>[125X [27X         if forders[i] mod 2 = 1 then[127X[104X
    [4X[25X>[125X [27X           pphi[i]:= 2 * cchi[i];[127X[104X
    [4X[25X>[125X [27X         elif cchi[i] = 0 then[127X[104X
    [4X[25X>[125X [27X           pphi[i]:= 0;[127X[104X
    [4X[25X>[125X [27X         fi;[127X[104X
    [4X[25X>[125X [27X         upper[i]:= 2* cchi[i];[127X[104X
    [4X[25X>[125X [27X       od;[127X[104X
    [4X[25X>[125X [27X       Append( chars, PermChars( f, rec( torso:= ShallowCopy( pphi ),[127X[104X
    [4X[25X>[125X [27X           upper:= upper,[127X[104X
    [4X[25X>[125X [27X           normalsubgroup:= [ 1 .. 4 ],[127X[104X
    [4X[25X>[125X [27X           nonfaithful:= cchi ) ) );[127X[104X
    [4X[25X>[125X [27X     od;[127X[104X
    [4X[25X>[125X [27X     Add( constit, List( Induced( Restricted( chars, g ), m ),[127X[104X
    [4X[25X>[125X [27X                         ValuesOfClassFunction ) );[127X[104X
    [4X[25X>[125X [27X   od;[127X[104X
    [4X[25Xgap>[125X [27XList( constit, Length );[127X[104X
    [4X[28X[ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 10, 10, 2 ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YNow we form the possible linear combinations.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xcand:= List( Cartesian( constit ), l -> sol * l );;[127X[104X
    [4X[25Xgap>[125X [27Xm:= CharacterTable( "M" );[127X[104X
    [4X[28XCharacterTable( "M" )[128X[104X
    [4X[25Xgap>[125X [27Xmorders:= OrdersClassRepresentatives( m );;[127X[104X
    [4X[25Xgap>[125X [27Xfor x in cand do[127X[104X
    [4X[25X>[125X [27X     for i in [ 1 .. Length( morders ) ] do[127X[104X
    [4X[25X>[125X [27X       if morders[i] mod 31 = 0 or morders[i] mod 21 = 0 then[127X[104X
    [4X[25X>[125X [27X         Unbind( x[i] );[127X[104X
    [4X[25X>[125X [27X       fi;[127X[104X
    [4X[25X>[125X [27X     od;[127X[104X
    [4X[25X>[125X [27X   od;[127X[104X
  [4X[32X[104X
  
  [33X[0;0YExactly one of the candidates has only integral values.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xcand:= Filtered( cand, x -> ForAll( x, IsInt ) );[127X[104X
    [4X[28X[ [ 391965121389536908413379198941796875, 23914487292951376996875, [128X[104X
    [4X[28X      474163138042468875, 9500455925885925, 646346515815, [128X[104X
    [4X[28X      334363486275, 954161764875, 147339103275, 1481392395, [128X[104X
    [4X[28X      1313281515, 0, 8203125, 9827885925, 1216215, 91556325, 9388791, [128X[104X
    [4X[28X      115911, 587331, 874650, 0, 79515, 581955, 336375, 104371, [128X[104X
    [4X[28X      62331, 36855, 0, 0, 0, 0, 28125, 525, 1125, 0, 188325, 16767, [128X[104X
    [4X[28X      88965, 2403, 9477, 1155, 891, 207, 351, 627, 0, 0, 4410, 1498, [128X[104X
    [4X[28X      0, 0, 0, 30, 150, 91, 151, 31, 0, 0, 0, 0, 0, 0, 0, 0, 0, 125, [128X[104X
    [4X[28X      0, 5, 5,,,,, 0, 0, 0, 0, 141, 45, 27, 61, 27, 9, 9, 7, 3, 15, [128X[104X
    [4X[28X      0, 0, 0, 0, 0, 98, 74, 42, 0, 0, 30, 0, 0, 0, 6, 6, 6,,, 1, 1, [128X[104X
    [4X[28X      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0,,,,, 0, [128X[104X
    [4X[28X      0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, [128X[104X
    [4X[28X      0, 0, 2, 2, 0, 2,,, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,,,, 0, [128X[104X
    [4X[28X      0, 0, 0, 0, 0,,, 0, 0, 0, 0, 0, 0,, 0, 0, 0 ] ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YNow  we compute the possible permutation characters that have the prescribed
  values.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xcand:= PermChars( m, rec( torso:= cand[1] ) );[127X[104X
    [4X[28X[ Character( CharacterTable( "M" ),[128X[104X
    [4X[28X  [ 391965121389536908413379198941796875, 23914487292951376996875, [128X[104X
    [4X[28X      474163138042468875, 9500455925885925, 646346515815, [128X[104X
    [4X[28X      334363486275, 954161764875, 147339103275, 1481392395, [128X[104X
    [4X[28X      1313281515, 0, 8203125, 9827885925, 1216215, 91556325, 9388791, [128X[104X
    [4X[28X      115911, 587331, 874650, 0, 79515, 581955, 336375, 104371, [128X[104X
    [4X[28X      62331, 36855, 0, 0, 0, 0, 28125, 525, 1125, 0, 188325, 16767, [128X[104X
    [4X[28X      88965, 2403, 9477, 1155, 891, 207, 351, 627, 0, 0, 4410, 1498, [128X[104X
    [4X[28X      0, 0, 0, 30, 150, 91, 151, 31, 0, 0, 0, 0, 0, 0, 0, 0, 0, 125, [128X[104X
    [4X[28X      0, 5, 5, 210, 0, 42, 0, 0, 0, 0, 0, 141, 45, 27, 61, 27, 9, 9, [128X[104X
    [4X[28X      7, 3, 15, 0, 0, 0, 0, 0, 98, 74, 42, 0, 0, 30, 0, 0, 0, 6, 6, [128X[104X
    [4X[28X      6, 3, 3, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, [128X[104X
    [4X[28X      1, 1, 0, 18, 0, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, [128X[104X
    [4X[28X      0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 2, 0, 2, 3, 3, 0, 0, 0, 0, [128X[104X
    [4X[28X      0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 2, 0, 0, 0, 0, 0, 0, 3, 3, 0, 0, [128X[104X
    [4X[28X      0, 0, 0, 0, 0, 0, 0, 0 ] ) ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YThere is only one candidate, so we have found the permutation character.[133X
  
  
  [1X8.16-4 [33X[0;0YThe Subgroup [22X2^{10+16}.O_10^+(2)[122X[101X[1X (November 2009)[133X[101X
  
  [33X[0;0YAccording  to the Atlas of Finite Groups [CCN+85, p. 234], the Monster group
  [22XM[122X  has  a  class  of  maximal  subgroups  [22XH[122X of the type [22X2^{10+16}.O_10^+(2)[122X.
  Currently  the  character  table  of  [22XH[122X  and the class fusion into [22XM[122X are not
  available  in [5XGAP[105X. We are interested in the permutation character [22X1_H^M[122X, and
  we will compute it without this information.[133X
  
  [33X[0;0YThe  subgroup [22XH[122X normalizes an elementary abelian group [22XE[122X of order [22X2^10[122X which
  contains  [22X496[122X  involutions  in the class [10X2A[110X and [22X527[122X involutions in the class
  [10X2B[110X. Let [22XV[122X denote the normal subgroup of order [22X2^26[122X in [22XH[122X, and set [22XbarH = H/N[122X.
  Since the smallest two indices of maximal subgroups of [22XbarH[122X are [22X496[122X and [22X527[122X,
  respectively,  [22XH[122X  acts transitively on both the [10X2A[110X and the [10X2B[110X involutions in
  [22XE[122X, and the centralizers of these involutions contain [22XV[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XHbar:= CharacterTable( "O10+(2)" );;[127X[104X
    [4X[25Xgap>[125X [27XU_Abar:= CharacterTable( "O10+(2)M1" );[127X[104X
    [4X[28XCharacterTable( "S8(2)" )[128X[104X
    [4X[25Xgap>[125X [27XIndex( Hbar, U_Abar );[127X[104X
    [4X[28X496[128X[104X
    [4X[25Xgap>[125X [27XU_Bbar:= CharacterTable( "O10+(2)M2" );[127X[104X
    [4X[28XCharacterTable( "2^8:O8+(2)" )[128X[104X
    [4X[25Xgap>[125X [27XIndex( Hbar, U_Bbar );[127X[104X
    [4X[28X527[128X[104X
  [4X[32X[104X
  
  [33X[0;0YWe  fix a [10X2A[110X involution [22Xz_A[122X in [22XE[122X, and set [22XG_A = C_M(z_A)[122X and [22XU_A = C_H(z_A)[122X.
  So  [22XG_A[122X  has  the  structure [22X2.B[122X and [22XU_A[122X has the structure [22X2^{10+16}.S_8(2)[122X.
  From  the  list of maximal subgroups of [22XB[122X we see that the image of [22XG_A[122X under
  the natural epimorphism from [22XG_A[122X to [22XB[122X is a maximal subgroup of [22XB[122X and has the
  structure [22X2^{9+16}.S_8(2)[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xb:= CharacterTable( "B" );[127X[104X
    [4X[28XCharacterTable( "B" )[128X[104X
    [4X[25Xgap>[125X [27XHorder:= 2^26 * Size( Hbar );[127X[104X
    [4X[28X1577011055923770163200[128X[104X
    [4X[25Xgap>[125X [27Xorder:= Horder / ( 2 * 496 );[127X[104X
    [4X[28X1589728887019929600[128X[104X
    [4X[25Xgap>[125X [27Xmaxes:= List( Maxes( b ), CharacterTable );;[127X[104X
    [4X[25Xgap>[125X [27Xfilt:= Filtered( maxes, t -> Size( t ) mod order = 0 );[127X[104X
    [4X[28X[ CharacterTable( "2^(9+16).S8(2)" ) ][128X[104X
    [4X[25Xgap>[125X [27XList( filt, t -> Size( t ) / order );[127X[104X
    [4X[28X[ 1 ][128X[104X
    [4X[25Xgap>[125X [27Xu1:= filt[1];[127X[104X
    [4X[28XCharacterTable( "2^(9+16).S8(2)" )[128X[104X
  [4X[32X[104X
  
  [33X[0;0YAnalogously, we fix a [10X2B[110X involution [22Xz_B[122X in [22XE[122X, and set [22XG_B = C_M(z_B)[122X and [22XU_B
  = C_H(z_B)[122X, Further, let [22XN[122X be the extraspecial normal subgroup of order [22X2^25[122X
  in  [22XG_B[122X.  So [22XG_B[122X has the structure [22X2^{1+24}_+.Co_1[122X, and [22XU_B[122X has index [22X527[122X in
  [22XG_B[122X. From the list of maximal subgroups of [22XCo_1[122X we see that the image of [22XU_B[122X
  under the natural epimorphism from [22XG_B[122X to [22XCo_1[122X is a maximal subgroup of [22XCo_1[122X
  and has the structure [22X2^{1+8}_+.O_8^+(2)[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xco1:= CharacterTable( "Co1" );;[127X[104X
    [4X[25Xgap>[125X [27Xorder:= Horder / ( 2^25 * 527 );[127X[104X
    [4X[28X89181388800[128X[104X
    [4X[25Xgap>[125X [27Xmaxes:= List( Maxes( co1 ), CharacterTable );;[127X[104X
    [4X[25Xgap>[125X [27Xfilt:= Filtered( maxes, t -> Size( t ) mod order = 0 );[127X[104X
    [4X[28X[ CharacterTable( "2^(1+8)+.O8+(2)" ) ][128X[104X
    [4X[25Xgap>[125X [27XList( filt, t -> Size( t ) / order );[127X[104X
    [4X[28X[ 1 ][128X[104X
    [4X[25Xgap>[125X [27Xu2:= filt[1];[127X[104X
    [4X[28XCharacterTable( "2^(1+8)+.O8+(2)" )[128X[104X
  [4X[32X[104X
  
  [33X[0;0YFirst  we  compute  the  permutation  characters  [22Xπ_A  = 1_{U_A}^M[122X and [22Xπ_B =
  1_{U_B}^M[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xm:= CharacterTable( "M" );[127X[104X
    [4X[28XCharacterTable( "M" )[128X[104X
    [4X[25Xgap>[125X [27X2b:= CharacterTable( "MC2A" );[127X[104X
    [4X[28XCharacterTable( "2.B" )[128X[104X
    [4X[25Xgap>[125X [27Xmm:= CharacterTable( "MC2B" );[127X[104X
    [4X[28XCharacterTable( "2^1+24.Co1" )[128X[104X
    [4X[25Xgap>[125X [27Xpi_A:= RestrictedClassFunction( TrivialCharacter( u1 )^b, 2b )^m;;[127X[104X
    [4X[25Xgap>[125X [27Xpi_B:= RestrictedClassFunction( TrivialCharacter( u2 )^co1, mm )^m;;[127X[104X
  [4X[32X[104X
  
  [33X[0;0YThe degree of [22X1_H^M[122X is of course known.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xtorso:= [ Size( m ) / Horder ];[127X[104X
    [4X[28X[ 512372707698741056749515292734375 ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YNext we compute some zero values of [22X1_H^M[122X, using the following conditions.[133X
  
  [30X    [33X[0;6YFor  [22Xg ∈ M[122X, if [22X|g|[122X does not divide [22X|H|[122X or if [22X|g|[122X is not the product of
        an  element order in [22XH/V[122X and a [22X2[122X-power. (In fact we could use that the
        exponent of [22XV[122X is [22X4[122X, but this would not improve the result.)[133X
  
  [30X    [33X[0;6YLet [22XU ≤ H ≤ G[122X, and let [22Xp[122X be a prime that does not divide [22X[H:U][122X. Then [22XU[122X
        contains  a  Sylow [22Xp[122X subgroup of [22XH[122X, so each element of order [22Xp[122X in [22XH[122X is
        conjugate  in  [22XH[122X  to  an element in [22XU[122X. For [22Xg ∈ G[122X, [22Xg = g_p h[122X, where the
        order  of  [22Xg_p[122X is a power of [22Xp[122X such that [22X1_U^G(g_p) = 0[122X holds, we have
        [22X1_H^G(g) = 0[122X. We apply this to [22XU ∈ { U_A, U_B }[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xmorders:= OrdersClassRepresentatives( m );;[127X[104X
    [4X[25Xgap>[125X [27X2parts:= Union( [ 1 ], Filtered( Set( morders ),[127X[104X
    [4X[25X>[125X [27X                         x -> IsPrimePowerInt( x ) and IsEvenInt( x ) ) );[127X[104X
    [4X[28X[ 1, 2, 4, 8, 16, 32 ][128X[104X
    [4X[25Xgap>[125X [27Xfactorders:= Set( OrdersClassRepresentatives( Hbar ) );;[127X[104X
    [4X[25Xgap>[125X [27Xprimes_A:= Filtered( PrimeDivisors( Horder ), p -> 496 mod p <> 0 );[127X[104X
    [4X[28X[ 3, 5, 7, 17 ][128X[104X
    [4X[25Xgap>[125X [27Xprimes_B:= Filtered( PrimeDivisors( Horder ), p -> 527 mod p <> 0 );[127X[104X
    [4X[28X[ 2, 3, 5, 7 ][128X[104X
    [4X[25Xgap>[125X [27Xprimes:= Union( primes_A, primes_B );;[127X[104X
    [4X[25Xgap>[125X [27Xn:= NrConjugacyClasses( m );;[127X[104X
    [4X[25Xgap>[125X [27Xfor i in [ 1 .. n ] do[127X[104X
    [4X[25X>[125X [27X  if Horder mod morders[i] <> 0 then[127X[104X
    [4X[25X>[125X [27X    torso[i]:= 0;[127X[104X
    [4X[25X>[125X [27X  elif ForAll( factorders, x -> not morders[i] / x in 2parts ) then[127X[104X
    [4X[25X>[125X [27X    torso[i]:= 0;[127X[104X
    [4X[25X>[125X [27X  else[127X[104X
    [4X[25X>[125X [27X    for p in primes do[127X[104X
    [4X[25X>[125X [27X      if morders[i] mod p = 0 then[127X[104X
    [4X[25X>[125X [27X        pprime:= morders[i];[127X[104X
    [4X[25X>[125X [27X        while pprime mod p = 0 do pprime:= pprime / p; od;[127X[104X
    [4X[25X>[125X [27X        pos:= PowerMap( m, pprime )[i];[127X[104X
    [4X[25X>[125X [27X        if p in primes_A and pi_A[ pos ] = 0 then[127X[104X
    [4X[25X>[125X [27X          torso[i]:= 0;[127X[104X
    [4X[25X>[125X [27X        elif p in primes_B and pi_B[ pos ] = 0 then[127X[104X
    [4X[25X>[125X [27X          torso[i]:= 0;[127X[104X
    [4X[25X>[125X [27X        fi;[127X[104X
    [4X[25X>[125X [27X      fi;[127X[104X
    [4X[25X>[125X [27X    od;[127X[104X
    [4X[25X>[125X [27X  fi;[127X[104X
    [4X[25X>[125X [27Xod;[127X[104X
    [4X[25Xgap>[125X [27Xtorso;[127X[104X
    [4X[28X[ 512372707698741056749515292734375,,,,, 0,,,,,,,,,,,, 0,, 0,,,,,,,,,,[128X[104X
    [4X[28X  ,,,, 0,,,, 0,,,,,, 0, 0, 0,,, 0,,,, 0,,,,,,,,,, 0,,,,,,,, 0, 0, 0, [128X[104X
    [4X[28X  0, 0, 0, 0,,,,, 0,,,,, 0, 0, 0, 0, 0, 0,,,, 0, 0,,,,, 0,,,,,,, 0, 0,[128X[104X
    [4X[28X  , 0, 0,,,,, 0, 0, 0, 0, 0,,,,, 0,, 0, 0, 0, 0, 0,, 0, 0, 0, 0, 0, 0,[128X[104X
    [4X[28X  , 0,, 0, 0, 0, 0,, 0, 0, 0, 0, 0,,,,,, 0,,, 0, 0,, 0, 0, 0, 0, 0, [128X[104X
    [4X[28X  0, 0, 0, 0,, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, [128X[104X
    [4X[28X  0, 0 ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YNow  we  want  to compute as many nonzero values of [22X1_H^M[122X as possible, using
  the  same  approach  as in the previous sections. For that, we first compute
  several  permutation characters [22X1_X^M[122X, for subgroups [22XX[122X with the property [22XV <
  X  <  U_A[122X  or [22XV < X < U_B[122X. Then we find several subsets [22XC[122X of [22XM[122X, each being a
  union  of  conjugacy  classes  of  [22XM[122X  such  that  [22X(1_H)_{C  ∩ H}[122X is a linear
  combination  of  the  characters [22X1_X^H[122X, restricted to [22XC ∩ H[122X. This yields the
  values of [22X1_H^M[122X on the classes in [22XC[122X.[133X
  
  [33X[0;0YThe  actual computations are performed with the characters [22X1_{X/V}^{H/V}[122X. So
  we  build two parallel lists [10Xcand[110X and [10Xcandbar[110X of permutation characters of [22XM[122X
  and of [22XH/V[122X, respectively. For that, we write two small [5XGAP[105X functions:[133X
  
  [30X    [33X[0;6YIn  the  function [10XAddSubgroupOfS82[110X, we choose a subgroup [22XY[122X of [22XS_8(2) ≅
        U_A/V[122X,  compute  [22X1_Y^{U_A/V}[122X, inflate it to a character of [22XU_A[122X, induce
        this  character  to  [22XB[122X,  inflate the result to [22XG_A[122X, and finally induce
        this character to [22XM[122X.[133X
  
  [30X    [33X[0;6YIn  the  function  [10XAddSubgroupOfO8p2[110X,  we  choose  a subgroup [22XY[122X of the
        factor  group  [22XF  ≅  O_8^+(2)[122X of [22XU_B/N[122X, compute [22X1_Y^F[122X, inflate it to a
        character  of  [22XU_B/N[122X,  induce  this  to  a  character of [22XG_B/N ≅ Co_1[122X,
        inflate  this to a character of [22XG_B[122X, and finally induce this character
        to [22XM[122X.[133X
  
        [33X[0;6YOne  difficulty  in  this  case is that choosing a subgroup [22XX/V[122X of [22XH/V[122X
        involves  fixing  the class fusion into [22XH/V[122X, but it is not clear which
        is  a  compatible class fusion of the corresponding subgroup [22XX[122X into [22XM[122X;
        therefore, each entry of [10Xcand[110X is in fact not the permutation character
        of [22XM[122X in question but a list of possibilities.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xcand:= [ [ pi_A ], [ pi_B ] ];;[127X[104X
    [4X[25Xgap>[125X [27Xcandbar:= [ TrivialCharacter( U_Abar )^Hbar,[127X[104X
    [4X[25X>[125X [27X               TrivialCharacter( U_Bbar )^Hbar ];;[127X[104X
    [4X[25Xgap>[125X [27XAddSubgroupOfS82:= function( subname )[127X[104X
    [4X[25X>[125X [27X  local psis82;[127X[104X
    [4X[25X>[125X [27X[127X[104X
    [4X[25X>[125X [27X  psis82:= TrivialCharacter( CharacterTable( subname ) )^U_Abar;[127X[104X
    [4X[25X>[125X [27X  Add( cand, [ Restricted( Restricted( psis82, u1 )^b, 2b )^m ] );[127X[104X
    [4X[25X>[125X [27X  Add( candbar, psis82 ^ Hbar );[127X[104X
    [4X[25X>[125X [27Xend;;[127X[104X
    [4X[25Xgap>[125X [27Xtt1:= CharacterTable( "O8+(2)" );[127X[104X
    [4X[28XCharacterTable( "O8+(2)" )[128X[104X
    [4X[25Xgap>[125X [27XAddSubgroupOfO8p2:= function( subname )[127X[104X
    [4X[25X>[125X [27X  local psi, list, char;[127X[104X
    [4X[25X>[125X [27X[127X[104X
    [4X[25X>[125X [27X  psi:= TrivialCharacter( CharacterTable( subname ) )^tt1;[127X[104X
    [4X[25X>[125X [27X  list:= [];[127X[104X
    [4X[25X>[125X [27X  for char in Orbit( AutomorphismsOfTable( tt1 ), psi, Permuted ) do[127X[104X
    [4X[25X>[125X [27X    AddSet( list, Restricted( Restricted( char, u2 ) ^ co1, mm ) ^ m );[127X[104X
    [4X[25X>[125X [27X  od;[127X[104X
    [4X[25X>[125X [27X  Add( cand, list );[127X[104X
    [4X[25X>[125X [27X  Add( candbar, Restricted( psi, U_Bbar ) ^ Hbar );[127X[104X
    [4X[25X>[125X [27Xend;;[127X[104X
  [4X[32X[104X
  
  [33X[0;0YNow  we  choose  the  subgroups  that will turn out to be sufficient for our
  computations.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XAddSubgroupOfS82( "O8+(2).2" );[127X[104X
    [4X[25Xgap>[125X [27XAddSubgroupOfO8p2( "S6(2)" );[127X[104X
    [4X[25Xgap>[125X [27XAddSubgroupOfS82( "O8-(2).2" );[127X[104X
    [4X[25Xgap>[125X [27XAddSubgroupOfS82( "A10.2" );[127X[104X
    [4X[25Xgap>[125X [27XAddSubgroupOfS82( "S4(4).2" );[127X[104X
    [4X[25Xgap>[125X [27XAddSubgroupOfS82( "L2(17)" );[127X[104X
    [4X[25Xgap>[125X [27XAddSubgroupOfO8p2( "A9" );[127X[104X
    [4X[25Xgap>[125X [27XAddSubgroupOfO8p2( "2^6:A8" );[127X[104X
    [4X[25Xgap>[125X [27XAddSubgroupOfO8p2( "(3xU4(2)):2" );[127X[104X
    [4X[25Xgap>[125X [27XAddSubgroupOfO8p2( "(A5xA5):2^2" );[127X[104X
    [4X[25Xgap>[125X [27XAddSubgroupOfS82( "S8(2)M4" );[127X[104X
  [4X[32X[104X
  
  [33X[0;0YIn  the  case  of  [22XA_5 < S_8(2)[122X, the function [10XAddSubgroupOfS82[110X does not work
  because  there  are  several class fusions of [22XA_5[122X into [22XS_8(2)[122X. We choose one
  fusion;  the  fact  that  it  really  describes  an embedding of an [22XA_5[122X type
  subgroup  of  [22XS_8(2)[122X can be checked using the function [2XNrPolyhedralSubgroups[102X
  ([14XReference: NrPolyhedralSubgroups[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xa5:= CharacterTable( "A5" );;[127X[104X
    [4X[25Xgap>[125X [27Xfus:= PossibleClassFusions( a5, U_Abar )[1];;[127X[104X
    [4X[25Xgap>[125X [27XNrPolyhedralSubgroups( U_Abar, fus[2], fus[3], fus[4] );[127X[104X
    [4X[28Xrec( number := 548352, type := "A5" )[128X[104X
    [4X[25Xgap>[125X [27Xpsis82:= Induced( a5, U_Abar, [ TrivialCharacter( a5 ) ], fus )[1];;[127X[104X
    [4X[25Xgap>[125X [27XAdd( cand, [ Restricted( Restricted( psis82, u1 )^b, 2b )^m ] );[127X[104X
    [4X[25Xgap>[125X [27XAdd( candbar, psis82 ^ Hbar );[127X[104X
    [4X[25Xgap>[125X [27XList( cand, Length );[127X[104X
    [4X[28X[ 1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 2, 2, 1, 1 ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  following  function  takes a condition on conjugacy classes in terms of
  their  element  orders,  which  gives a set [22XC[122X of elements in [22XM[122X. It forms the
  corresponding set of elements in [22XH/V[122X and tries to express the restriction of
  [22X1_{H/V}[122X  as a linear combination of the characters [22X1_X^{H/V}[122X that are stored
  in  the  list  [10Xcandbar[110X.  If  this  works  and  if  the  corresponding linear
  combination  of  the candidates in [10Xcand[110X is unique, the newly found values of
  [22X1_H^M[122X are entered into the list [10Xtorso[110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XHbarorders:= OrdersClassRepresentatives( Hbar );;[127X[104X
    [4X[25Xgap>[125X [27XTryCondition:= function( cond )[127X[104X
    [4X[25X>[125X [27X  local pos, sol, lincomb, oldknown, i;[127X[104X
    [4X[25X>[125X [27X[127X[104X
    [4X[25X>[125X [27X  pos:= Filtered( [ 1 .. Length( Hbarorders ) ],[127X[104X
    [4X[25X>[125X [27X            i -> cond( Hbarorders[i] ) );[127X[104X
    [4X[25X>[125X [27X  sol:= SolutionMat( candbar{[1..Length(candbar)]}{ pos },[127X[104X
    [4X[25X>[125X [27X            ListWithIdenticalEntries( Length( pos ), 1 ) );[127X[104X
    [4X[25X>[125X [27X  if sol = fail then[127X[104X
    [4X[25X>[125X [27X    return "no solution";[127X[104X
    [4X[25X>[125X [27X  fi;[127X[104X
    [4X[25X>[125X [27X[127X[104X
    [4X[25X>[125X [27X  pos:= Filtered( [ 1 .. Length( morders) ], i -> cond( morders[i] ) );[127X[104X
    [4X[25X>[125X [27X  lincomb:= Filtered( Set( Cartesian( cand ), x -> sol * x ),[127X[104X
    [4X[25X>[125X [27X                x -> ForAll( pos, i -> IsPosInt( x[i] ) or x[i] = 0 ) );[127X[104X
    [4X[25X>[125X [27X  if Length( lincomb ) <> 1 then[127X[104X
    [4X[25X>[125X [27X    return "solution is not unique";[127X[104X
    [4X[25X>[125X [27X  fi;[127X[104X
    [4X[25X>[125X [27X[127X[104X
    [4X[25X>[125X [27X  lincomb:= lincomb[1];;[127X[104X
    [4X[25X>[125X [27X  oldknown:= Number( torso );[127X[104X
    [4X[25X>[125X [27X  for i in pos do[127X[104X
    [4X[25X>[125X [27X    if IsBound( torso[i] ) then[127X[104X
    [4X[25X>[125X [27X      if torso[i] <> lincomb[i] then[127X[104X
    [4X[25X>[125X [27X        Error( "contradiction of new and known value at position ", i );[127X[104X
    [4X[25X>[125X [27X      fi;[127X[104X
    [4X[25X>[125X [27X    elif not IsInt( lincomb[i] ) or lincomb[i] < 0 then[127X[104X
    [4X[25X>[125X [27X      Error( "new value at position ", i, " is not a nonneg. integer" );[127X[104X
    [4X[25X>[125X [27X    fi;[127X[104X
    [4X[25X>[125X [27X    torso[i]:= lincomb[i];[127X[104X
    [4X[25X>[125X [27X  od;[127X[104X
    [4X[25X>[125X [27X  return Concatenation( "now ", String( Number( torso ) ), " values (",[127X[104X
    [4X[25X>[125X [27X             String( Number( torso ) - oldknown ), " new)" );[127X[104X
    [4X[25X>[125X [27Xend;;[127X[104X
  [4X[32X[104X
  
  [33X[0;0YThis  procedure  makes  sense  only  if  the  elements of [22XH[122X that satisfy the
  condition  are  contained  in  the  full preimage of the classes of [22XH/V[122X that
  satisfy the condition. Note that this is in fact the case for the conditions
  used  below. This is clear for condition concerning only [13Xodd[113X element orders,
  because  [22XV[122X  is  a [22X2[122X-group. Also the set of all elements of the orders [22X9[122X, [22X18[122X,
  and  [22X36[122X  is such a [21Xclosed[121X set, since [22XM[122X has no elements of order [22X72[122X. Finally,
  the set of all elements of the orders [22X1[122X, [22X2[122X, and [22X4[122X in [22XH[122X is admissible because
  it  is  contained in the preimage of the set of all elements of these orders
  in [22XH/V[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XTryCondition( x -> x mod 7 = 0 and x mod 3 <> 0 );[127X[104X
    [4X[28X"now 99 values (7 new)"[128X[104X
    [4X[25Xgap>[125X [27XTryCondition( x -> x mod 17 = 0 and x mod 3 <> 0 );[127X[104X
    [4X[28X"now 102 values (3 new)"[128X[104X
    [4X[25Xgap>[125X [27XTryCondition( x -> x mod 5 = 0 and x mod 3 <> 0 );[127X[104X
    [4X[28X"now 119 values (17 new)"[128X[104X
    [4X[25Xgap>[125X [27XTryCondition( x -> 4 mod x = 0 );[127X[104X
    [4X[28X"now 125 values (6 new)"[128X[104X
    [4X[25Xgap>[125X [27XTryCondition( x -> 9 mod x = 0 );[127X[104X
    [4X[28X"now 129 values (4 new)"[128X[104X
    [4X[25Xgap>[125X [27XTryCondition( x -> x in [ 9, 18, 36 ] );[127X[104X
    [4X[28X"now 138 values (9 new)"[128X[104X
  [4X[32X[104X
  
  [33X[0;0YPossible  constituents of [22X1_H^M[122X are those rational irreducible characters of
  [22XM[122X that are constituents of [22Xπ^M[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xconstit:= Filtered( RationalizedMat( Irr( m ) ),[127X[104X
    [4X[25X>[125X [27X              x -> ScalarProduct( m, x, pi_A ) <> 0[127X[104X
    [4X[25X>[125X [27X                   and ScalarProduct( m, x, pi_B ) <> 0 );;[127X[104X
  [4X[32X[104X
  
  [33X[0;0YFor the missing values, we can provide at least lower bounds, using that [22XU ≤
  H ≤ G[122X implies [22X1_H^G(g) ≥ 1_U^G(g) / [H:U] = [G:H] ⋅ 1_U^G(g) / 1_U^G(1)[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xlower:= [];;[127X[104X
    [4X[25Xgap>[125X [27XHindex:= Size( m ) / Horder;[127X[104X
    [4X[28X512372707698741056749515292734375[128X[104X
    [4X[25Xgap>[125X [27Xfor i in [ 1 .. NrConjugacyClasses( m ) ] do[127X[104X
    [4X[25X>[125X [27X  lower[i]:= Maximum( pi_A[i] / ( pi_A[1] / Hindex ),[127X[104X
    [4X[25X>[125X [27X                      pi_B[i] / ( pi_B[1] / Hindex ) );[127X[104X
    [4X[25X>[125X [27X  if not IsInt( lower[i] ) then[127X[104X
    [4X[25X>[125X [27X    lower[i]:= Int( lower[i] + 1 );[127X[104X
    [4X[25X>[125X [27X  fi;[127X[104X
    [4X[25X>[125X [27Xod;[127X[104X
  [4X[32X[104X
  
  [33X[0;0YNow  we compute the possible permutation characters that have the prescribed
  values, are compatible with the given lower bounds for values, and have only
  constituents in the given list.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xcand:= PermChars( m, rec( torso:= torso, chars:= constit,[127X[104X
    [4X[25X>[125X [27X     lower:= lower, normalsubgroup:= [ 1 .. NrConjugacyClasses( m ) ],[127X[104X
    [4X[25X>[125X [27X     nonfaithful:= TrivialCharacter( m ) ) );[127X[104X
    [4X[28X[ Character( CharacterTable( "M" ),[128X[104X
    [4X[28X  [ 512372707698741056749515292734375, 405589064025344574375, [128X[104X
    [4X[28X      29628786742129575, 658201521662685, 215448838605, 0, [128X[104X
    [4X[28X      121971774375, 28098354375, 335229607, 108472455, 164587500, [128X[104X
    [4X[28X      4921875, 2487507165, 2567565, 26157789, 6593805, 398925, 0, [128X[104X
    [4X[28X      437325, 0, 44983, 234399, 90675, 21391, 41111, 12915, 6561, [128X[104X
    [4X[28X      6561, 177100, 7660, 6875, 315, 275, 0, 113373, 17901, 57213, 0, [128X[104X
    [4X[28X      4957, 1197, 909, 301, 397, 0, 0, 0, 3885, 525, 0, 2835, 90, 45, [128X[104X
    [4X[28X      0, 103, 67, 43, 28, 81, 189, 9, 9, 9, 0, 540, 300, 175, 20, 15, [128X[104X
    [4X[28X      7, 420, 0, 0, 0, 0, 0, 0, 0, 165, 61, 37, 37, 0, 9, 9, 13, 5, [128X[104X
    [4X[28X      0, 0, 0, 0, 0, 0, 77, 45, 13, 0, 0, 45, 115, 19, 10, 0, 5, 5, [128X[104X
    [4X[28X      9, 9, 1, 1, 0, 0, 4, 0, 0, 9, 9, 3, 1, 0, 0, 0, 0, 0, 0, 4, 1, [128X[104X
    [4X[28X      1, 0, 24, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 1, 0, 4, 0, 0, 0, [128X[104X
    [4X[28X      0, 1, 0, 0, 0, 0, 0, 3, 3, 1, 1, 2, 0, 3, 3, 0, 0, 0, 0, 0, 0, [128X[104X
    [4X[28X      0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, [128X[104X
    [4X[28X      0, 0, 0, 0, 0, 0 ] ) ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YThere is only one candidate, so we have found the permutation character.[133X
  
  
  [1X8.17 [33X[0;0YA permutation character of the Baby Monster (June 2012)[133X[101X
  
  [33X[0;0YWe  compute  the  character  of  the  Baby  Monster that is induced from the
  trivial  character  of a Sylow [22X2[122X-subgroup. (Gabriel Navarro had asked me how
  [5XGAP[105X  can  compute  this  character.)  We start with the computation of those
  transitive permutation characters of the symmetric group on five points that
  have  degree  [22X15[122X.  Note  that  the function [2XPermChars[102X ([14XReference: PermChars[114X)
  computes  in  general  only candidates, but here we are sure that the result
  consists of permutation characters because it is unique.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xt:= CharacterTable( "S5" );[127X[104X
    [4X[28XCharacterTable( "A5.2" )[128X[104X
    [4X[25Xgap>[125X [27Xpi:= PermChars( t, rec( torso:= [ 15 ] ) );[127X[104X
    [4X[28X[ Character( CharacterTable( "A5.2" ), [ 15, 3, 0, 0, 3, 1, 0 ] ) ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YNext,  we  regard  this  character  as a character of the group [22X2^5:S_5[122X that
  occurs as a maximal subgroup of index [22X231[122X in [22XM_22:2[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xm222:= CharacterTable( "M22.2" );[127X[104X
    [4X[28XCharacterTable( "M22.2" )[128X[104X
    [4X[25Xgap>[125X [27Xmx:= List( Maxes( m222 ), CharacterTable );;[127X[104X
    [4X[25Xgap>[125X [27Xmx:= Filtered( mx, x -> Size( m222 ) / Size( x ) = 231 );[127X[104X
    [4X[28X[ CharacterTable( "M22.2M4" ) ][128X[104X
    [4X[25Xgap>[125X [27Xpi:= pi[1]{ GetFusionMap( mx[1], t ) };[127X[104X
    [4X[28X[ 15, 15, 3, 3, 3, 0, 0, 3, 3, 1, 1, 0, 15, 15, 3, 3, 3, 0, 0, 3, 3, [128X[104X
    [4X[28X  1, 1, 0 ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YWe induce this character to [22XM_22:2[122X. (Note that this is the character that is
  induced from the trivial character of a Sylow [22X2[122X-subgroup of [22XM_22:2[122X.)[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xpi:= InducedClassFunction( mx[1], pi, m222 );[127X[104X
    [4X[28XClassFunction( CharacterTable( "M22.2" ),[128X[104X
    [4X[28X [ 3465, 105, 0, 9, 5, 0, 0, 0, 0, 1, 0, 189, 45, 9, 13, 0, 1, 0, 0, [128X[104X
    [4X[28X  0, 0 ] )[128X[104X
  [4X[32X[104X
  
  [33X[0;0YNext,  we regard this character as a character of the group [22X2^10:M_22:2[122X that
  occurs as a maximal subgroup of index [22X46575[122X in [22XCo_2[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xco2:= CharacterTable( "Co2" );[127X[104X
    [4X[28XCharacterTable( "Co2" )[128X[104X
    [4X[25Xgap>[125X [27Xmx:= List( Maxes( co2 ), CharacterTable );;[127X[104X
    [4X[25Xgap>[125X [27Xmx:= Filtered( mx, x -> Size( co2 ) / Size( x ) = 46575 );[127X[104X
    [4X[28X[ CharacterTable( "2^10:m22:2" ) ][128X[104X
    [4X[25Xgap>[125X [27Xpi:= pi{ GetFusionMap( mx[1], m222 ) };[127X[104X
    [4X[28X[ 3465, 3465, 3465, 3465, 105, 105, 105, 105, 105, 105, 105, 105, 0, [128X[104X
    [4X[28X  0, 0, 0, 0, 9, 9, 9, 9, 9, 9, 5, 5, 5, 5, 5, 0, 0, 0, 0, 0, 0, 0, [128X[104X
    [4X[28X  0, 0, 0, 1, 1, 1, 0, 189, 189, 189, 189, 189, 189, 45, 45, 45, 45, [128X[104X
    [4X[28X  9, 9, 9, 9, 13, 13, 13, 13, 13, 13, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, [128X[104X
    [4X[28X  0, 0, 0, 0, 0, 0, 0 ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YWe induce this character to [22XCo_2[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xpi:= InducedClassFunction( mx[1], pi, co2 );[127X[104X
    [4X[28XClassFunction( CharacterTable( "Co2" ),[128X[104X
    [4X[28X [ 161382375, 626535, 162855, 27495, 0, 0, 6615, 3975, 2727, 855, [128X[104X
    [4X[28X  567, 975, 115, 0, 0, 0, 0, 0, 0, 0, 0, 0, 63, 51, 19, 27, 35, 7, 0, [128X[104X
    [4X[28X  0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, [128X[104X
    [4X[28X  0, 0, 0, 0, 0, 0, 0, 0, 0 ] )[128X[104X
  [4X[32X[104X
  
  [33X[0;0YNext,  we  regard  this  character as a character of the group [22X2^{1+22}.Co_2[122X
  that  occurs  as  a  maximal  subgroup  of  index [22X11707448673375[122X in the Baby
  Monster.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xb:= CharacterTable( "B" );[127X[104X
    [4X[28XCharacterTable( "B" )[128X[104X
    [4X[25Xgap>[125X [27Xmx:= List( Maxes( b ), CharacterTable );;[127X[104X
    [4X[25Xgap>[125X [27Xmx:= Filtered( mx, x -> Size( b ) / Size( x ) = 11707448673375 );[127X[104X
    [4X[28X[ CharacterTable( "2^(1+22).Co2" ) ][128X[104X
    [4X[25Xgap>[125X [27Xpi:= pi{ GetFusionMap( mx[1], co2 ) };;[127X[104X
    [4X[25Xgap>[125X [27Xpi[1];[127X[104X
    [4X[28X161382375[128X[104X
  [4X[32X[104X
  
  [33X[0;0YWe induce this character to the Baby Monster.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xpi:= InducedClassFunction( mx[1], pi, b );[127X[104X
    [4X[28XClassFunction( CharacterTable( "B" ),[128X[104X
    [4X[28X [ 1889375872099856765625, 2609385408855225, 62316674429625, [128X[104X
    [4X[28X  207818526825, 268788490425, 0, 0, 13052741625, 7537207545, [128X[104X
    [4X[28X  128298681, 270580905, 46366425, 74315385, 35633385, 3937689, [128X[104X
    [4X[28X  201825, 1233225, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 713097, [128X[104X
    [4X[28X  241425, 320625, 88521, 275265, 57705, 19305, 20089, 9441, 6489, [128X[104X
    [4X[28X  2577, 1825, 5345, 753, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, [128X[104X
    [4X[28X  0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, [128X[104X
    [4X[28X  273, 417, 105, 97, 185, 33, 9, 9, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, [128X[104X
    [4X[28X  0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, [128X[104X
    [4X[28X  0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, [128X[104X
    [4X[28X  0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, [128X[104X
    [4X[28X  0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ] )[128X[104X
  [4X[32X[104X
  
  
  [1X8.18 [33X[0;0YA permutation character of [22X2.B[122X[101X[1X (October 2017)[133X[101X
  
  [33X[0;0YWe compute the character of the double cover [22X2.B[122X of the Baby Monster that is
  induced  from  the  trivial  character  of  a  subgroup  [22XU[122X  of the structure
  [22X2^1+22.McL[122X.[133X
  
  [33X[0;0YThis subgroup occurs as the intersection of two conjugates of [22X2.B[122X inside the
  Monster  group  [22XM[122X.  More precisely, we consider [22X2.B[122X as the centralizer of an
  involution  [22Xa[122X  in [22XM[122X, and we are interested in the permutation action of [22XM[122X on
  the  cosets  of  [22X2.B[122X  (or, equivalently, on the conjugacy class in [22XM[122X of this
  involution).  The  restriction of this action to [22X2.B[122X has nine orbits. One of
  them has point stabilizer [22XU[122X.[133X
  
  [33X[0;0YBackground  information can be found in [GJMS89]. The decomposition into the
  nine  orbits  appears  in  Definition  (3.4.9)  on  p 587,  and our orbit is
  characterized  in  Table VII (on p. 582) by the facts that its points [22Xc[122X have
  order [22X4[122X and the squares of [22Xa c[122X lie in the class [10X2B[110X of [22XM[122X. This implies that [22Xa[122X
  and [22Xc[122X do not commute, hence [22Xa[122X does not lie in [22XU[122X.[133X
  
  [33X[0;0YFrom this description, we know that [22XU[122X is a subgroup of a maximal subgroup of
  the type [22X2^2+22.Co_2[122X in [22X2.B[122X, and the group [22X⟨ U, a ⟩[122X has the type [22X2^2+22.McL[122X.[133X
  
  [33X[0;0YThus we can proceed in two steps. First we induce the trivial character of [22X⟨
  U,  a  ⟩[122X  to  [22X2.B[122X.  Then  we  use  the variant of the [5XGAP[105X function [2XPermChars[102X
  ([14XReference: PermChars[114X) that allows us to prescribe the permutation character
  of the closure with a normal subgroup, which is [22X⟨ a ⟩[122X in our case.[133X
  
  [33X[0;0YThe  first step can be performed by inducing the trivial character of [22XMcL[122X to
  [22XCo_2[122X, [22X...[122X[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xmcl:= CharacterTable( "McL" );[127X[104X
    [4X[28XCharacterTable( "McL" )[128X[104X
    [4X[25Xgap>[125X [27Xco2:= CharacterTable( "Co2" );[127X[104X
    [4X[28XCharacterTable( "Co2" )[128X[104X
    [4X[25Xgap>[125X [27Xind:= Induced( mcl, co2, [ TrivialCharacter( mcl ) ] )[1];[127X[104X
    [4X[28XCharacter( CharacterTable( "Co2" ),[128X[104X
    [4X[28X [ 47104, 0, 1024, 0, 16, 160, 0, 0, 0, 0, 64, 0, 0, 4, 24, 16, 0, 0, [128X[104X
    [4X[28X  0, 16, 0, 8, 0, 0, 0, 0, 0, 8, 4, 4, 0, 0, 2, 0, 0, 0, 0, 0, 0, 4, [128X[104X
    [4X[28X  0, 0, 2, 2, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1 ] )[128X[104X
  [4X[32X[104X
  
  [33X[0;0Y[22X...[122X regarding this character as a character of [22X2^1+22.Co_2[122X, [22X...[122X[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xm:= CharacterTable( "BM2" );[127X[104X
    [4X[28XCharacterTable( "2^(1+22).Co2" )[128X[104X
    [4X[25Xgap>[125X [27Xinfl:= ind{ GetFusionMap( m, co2 ) };[127X[104X
    [4X[28X[ 47104, 47104, 47104, 47104, 47104, 47104, 47104, 0, 0, 0, 0, 0, 0, [128X[104X
    [4X[28X  0, 0, 0, 0, 0, 0, 1024, 1024, 1024, 1024, 1024, 1024, 1024, 1024, [128X[104X
    [4X[28X  1024, 1024, 1024, 1024, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, [128X[104X
    [4X[28X  0, 0, 0, 0, 16, 16, 16, 16, 160, 160, 160, 160, 160, 160, 160, 160, [128X[104X
    [4X[28X  160, 160, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, [128X[104X
    [4X[28X  0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, [128X[104X
    [4X[28X  0, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 0, 0, 0, [128X[104X
    [4X[28X  0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, [128X[104X
    [4X[28X  0, 0, 0, 0, 4, 4, 4, 24, 24, 24, 24, 24, 24, 24, 24, 16, 16, 16, [128X[104X
    [4X[28X  16, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, [128X[104X
    [4X[28X  0, 0, 0, 0, 0, 0, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, [128X[104X
    [4X[28X  16, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8, 8, 8, 8, [128X[104X
    [4X[28X  8, 8, 8, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, [128X[104X
    [4X[28X  0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, [128X[104X
    [4X[28X  0, 0, 0, 0, 0, 0, 0, 8, 8, 8, 8, 8, 8, 8, 8, 8, 4, 4, 4, 4, 4, 4, [128X[104X
    [4X[28X  4, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, [128X[104X
    [4X[28X  2, 2, 2, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, [128X[104X
    [4X[28X  0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 4, 4, [128X[104X
    [4X[28X  4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, [128X[104X
    [4X[28X  2, 2, 2, 2, 2, 2, 2, 2, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, [128X[104X
    [4X[28X  0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, [128X[104X
    [4X[28X  0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, [128X[104X
    [4X[28X  1, 1, 1, 1 ][128X[104X
  [4X[32X[104X
  
  [33X[0;0Y[22X...[122X inducing this character to [22XB[122X, [22X...[122X[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xb:= CharacterTable( "B" );[127X[104X
    [4X[28XCharacterTable( "B" )[128X[104X
    [4X[25Xgap>[125X [27Xind:= Induced( m, b, [ infl ] )[1];[127X[104X
    [4X[28XClassFunction( CharacterTable( "B" ),[128X[104X
    [4X[28X [ 551467662310656000, 186911262720, 272993634304, 0, 634521600, [128X[104X
    [4X[28X  194594400, 69984, 8495104, 17465344, 129024, 276480, 2073600, [128X[104X
    [4X[28X  16384, 798720, 46080, 0, 5120, 138600, 1000, 110880, 252000, [128X[104X
    [4X[28X  112480, 432, 12960, 0, 1312, 8352, 864, 432, 0, 2520, 0, 2880, [128X[104X
    [4X[28X  2880, 3072, 2880, 0, 0, 256, 64, 1152, 576, 640, 192, 96, 0, 108, [128X[104X
    [4X[28X  2520, 744, 0, 104, 120, 40, 30, 160, 16, 1120, 1024, 0, 0, 96, 288, [128X[104X
    [4X[28X  64, 144, 0, 96, 0, 108, 16, 48, 0, 32, 12, 0, 0, 0, 168, 0, 104, [128X[104X
    [4X[28X  48, 0, 4, 0, 0, 0, 0, 32, 16, 8, 8, 0, 24, 12, 4, 0, 0, 0, 0, 24, [128X[104X
    [4X[28X  4, 24, 24, 0, 0, 0, 0, 4, 0, 0, 6, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, [128X[104X
    [4X[28X  0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 8, 0, 16, 8, 4, 0, 0, 0, 0, 0, 4, 2, [128X[104X
    [4X[28X  2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, [128X[104X
    [4X[28X  0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ] )[128X[104X
  [4X[32X[104X
  
  [33X[0;0Y[22X...[122X and regarding the result as a character of [22X2.B[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27X2b:= CharacterTable( "2.B" );[127X[104X
    [4X[28XCharacterTable( "2.B" )[128X[104X
    [4X[25Xgap>[125X [27Xinfl:= ind{ GetFusionMap( 2b, b ) };[127X[104X
    [4X[28X[ 551467662310656000, 551467662310656000, 186911262720, 272993634304, [128X[104X
    [4X[28X  272993634304, 0, 634521600, 194594400, 194594400, 69984, 69984, [128X[104X
    [4X[28X  8495104, 17465344, 129024, 276480, 2073600, 2073600, 16384, 798720, [128X[104X
    [4X[28X  46080, 0, 5120, 138600, 138600, 1000, 1000, 110880, 252000, 112480, [128X[104X
    [4X[28X  112480, 432, 12960, 0, 1312, 1312, 8352, 864, 864, 432, 0, 2520, [128X[104X
    [4X[28X  2520, 0, 2880, 2880, 3072, 2880, 0, 0, 256, 64, 1152, 576, 576, [128X[104X
    [4X[28X  640, 192, 96, 0, 0, 108, 108, 2520, 744, 744, 0, 104, 104, 120, 40, [128X[104X
    [4X[28X  40, 30, 30, 160, 16, 1120, 1024, 0, 0, 0, 96, 288, 64, 144, 144, 0, [128X[104X
    [4X[28X  96, 0, 108, 108, 16, 48, 0, 32, 12, 12, 0, 0, 0, 0, 168, 0, 104, [128X[104X
    [4X[28X  104, 48, 0, 0, 4, 4, 0, 0, 0, 0, 32, 16, 8, 8, 8, 0, 0, 24, 12, 4, [128X[104X
    [4X[28X  4, 0, 0, 0, 0, 0, 0, 24, 4, 24, 24, 0, 0, 0, 0, 0, 4, 0, 0, 0, 6, [128X[104X
    [4X[28X  6, 6, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, [128X[104X
    [4X[28X  0, 0, 0, 0, 0, 0, 8, 0, 16, 8, 4, 4, 0, 0, 0, 0, 0, 0, 4, 4, 2, 2, [128X[104X
    [4X[28X  2, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, [128X[104X
    [4X[28X  0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 0, 0, 0, 0, 0, 0, 0, 0, [128X[104X
    [4X[28X  0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YNow  we  have  the  character  [22Xψ[122X that represents the [21Xnonfaithful half[121X of the
  desired  permutation  character.  We  have  to  [21Xfill  it  up[121X  with  faithful
  characters  of  [22X2.B[122X  of  total degree [22Xψ(1)[122X such that the sum with [22Xψ[122X can be a
  permutation character of [22X2.B[122X.[133X
  
  [33X[0;0YThe  [5XGAP[105X  function  [2XPermChars[102X  ([14XReference:  PermChars[114X)  is designed for this
  situation. We specify the normal subgroup [22XN = ⟨ a ⟩[122X by listing the positions
  of  its  conjugacy classes in the character table of [22X2.B[122X, we enter the known
  permutation  character  [22X1_{UN}^{2.B}[122X, and of course we specify the degree of
  the possible permutation characters.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xcentre:= ClassPositionsOfCentre( 2b );[127X[104X
    [4X[28X[ 1, 2 ][128X[104X
    [4X[25Xgap>[125X [27Xpi:= PermChars( 2b, rec( torso:= [ 2 * infl[1], 0 ],[127X[104X
    [4X[25X>[125X [27X                            normalsubgroup:= centre,[127X[104X
    [4X[25X>[125X [27X                            nonfaithful:= infl ) );[127X[104X
    [4X[28X[ Character( CharacterTable( "2.B" ),[128X[104X
    [4X[28X  [ 1102935324621312000, 0, 186911262720, 541790208000, 4197060608, [128X[104X
    [4X[28X      0, 634521600, 389188800, 0, 139968, 0, 8495104, 17465344, [128X[104X
    [4X[28X      129024, 276480, 4026240, 120960, 16384, 798720, 46080, 0, 5120, [128X[104X
    [4X[28X      277200, 0, 2000, 0, 110880, 252000, 190080, 34880, 432, 12960, [128X[104X
    [4X[28X      0, 2592, 32, 8352, 1728, 0, 432, 0, 5040, 0, 0, 2880, 2880, [128X[104X
    [4X[28X      3072, 2880, 0, 0, 256, 64, 1152, 1008, 144, 640, 192, 96, 0, 0, [128X[104X
    [4X[28X      216, 0, 2520, 960, 528, 0, 200, 8, 120, 80, 0, 60, 0, 160, 16, [128X[104X
    [4X[28X      1120, 1024, 0, 0, 0, 96, 288, 64, 216, 72, 0, 96, 0, 216, 0, [128X[104X
    [4X[28X      16, 48, 0, 32, 24, 0, 0, 0, 0, 0, 168, 0, 160, 48, 48, 0, 0, 8, [128X[104X
    [4X[28X      0, 0, 0, 0, 0, 32, 16, 8, 12, 4, 0, 0, 24, 12, 0, 8, 0, 0, 0, [128X[104X
    [4X[28X      0, 0, 0, 24, 4, 24, 24, 0, 0, 0, 0, 0, 4, 0, 0, 0, 6, 6, 8, 4, [128X[104X
    [4X[28X      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, [128X[104X
    [4X[28X      0, 0, 0, 0, 8, 0, 16, 8, 8, 0, 0, 0, 0, 0, 0, 0, 8, 0, 2, 2, 2, [128X[104X
    [4X[28X      2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, [128X[104X
    [4X[28X      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 0, 0, 0, 0, 0, 0, 0, [128X[104X
    [4X[28X      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ] ) ][128X[104X
    [4X[25Xgap>[125X [27XMatScalarProducts( 2b, Irr( 2b ), pi );[127X[104X
    [4X[28X[ [ 1, 1, 2, 1, 2, 0, 2, 3, 2, 0, 0, 1, 4, 1, 2, 0, 3, 2, 0, 2, 0, 0, [128X[104X
    [4X[28X      2, 2, 0, 0, 2, 3, 1, 5, 0, 4, 3, 2, 0, 0, 3, 2, 0, 6, 4, 0, 1, [128X[104X
    [4X[28X      1, 0, 0, 0, 0, 3, 0, 1, 0, 0, 5, 0, 5, 2, 0, 0, 2, 0, 0, 4, 1, [128X[104X
    [4X[28X      0, 2, 0, 4, 2, 4, 4, 3, 0, 2, 4, 2, 4, 0, 3, 0, 3, 2, 5, 0, 1, [128X[104X
    [4X[28X      0, 3, 1, 0, 1, 1, 2, 5, 3, 1, 1, 4, 5, 1, 1, 0, 3, 0, 0, 3, 2, [128X[104X
    [4X[28X      1, 1, 2, 1, 1, 4, 0, 3, 2, 3, 1, 3, 0, 1, 3, 0, 2, 2, 1, 3, 3, [128X[104X
    [4X[28X      0, 0, 2, 0, 0, 0, 0, 3, 0, 3, 3, 3, 1, 0, 3, 0, 4, 0, 1, 0, 0, [128X[104X
    [4X[28X      2, 0, 0, 2, 0, 0, 2, 1, 1, 0, 0, 0, 0, 1, 2, 1, 1, 1, 0, 1, 1, [128X[104X
    [4X[28X      1, 1, 1, 1, 0, 2, 1, 1, 3, 3, 0, 0, 0, 1, 1, 1, 1, 2, 3, 2, 0, [128X[104X
    [4X[28X      0, 2, 2, 4, 3, 5, 2, 4, 0, 0, 0, 0, 5, 2, 0, 0, 0, 1, 1, 0, 0, [128X[104X
    [4X[28X      0, 0, 0, 0, 7, 0, 0, 1, 7, 7, 0, 0, 0, 1, 6, 4, 5, 0, 0, 3, 0, [128X[104X
    [4X[28X      0, 0, 0, 0, 4, 1, 1, 3, 8, 3, 2, 2, 5, 0, 1 ] ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YWe are lucky: There is a unique solution, and its computation is quite fast.[133X
  
  
  [1X8.19   [33X[0;0YGeneration   of   sporadic  simple  groups  by  [22Xπ[122X[101X[1X-  and  [22Xπ'[122X[101X[1X-subgroups[101X
  [1X(December 2021)[133X[101X
  
  [33X[0;0YThis  section  shows  the  computations that are needed in order to show the
  following statements from [BG].[133X
  
  [33X[0;0Y[13XProposition 2.2[113X:  Let  [22XS[122X  be  a  sporadic  simple group and let [22XP[122X be a Sylow
  [22X2[122X-subgroup of [22XS[122X. If [22X1 ≠ x ∈ S[122X, then [22XS = ⟨ P, x^g ⟩[122X for some [22Xg ∈ S[122X.[133X
  
  [33X[0;0Y[13XTheorem 7.1[113X:  Let  [22XS[122X be a sporadic simple group and let [22Xp ≤ q[122X be primes each
  dividing  [22X|S|[122X.  Then  [22XS[122X  can  be generated by a Sylow [22Xp[122X-subgroup and a Sylow
  [22Xq[122X-subgroup.[133X
  
  [33X[0;0YA  stronger  version of Theorem 7.1: Let [22XS[122X be a sporadic simple group [22Xp[122X be a
  prime dividing [22X|S|[122X, and [22XP[122X be a Sylow [22Xp[122X-subgroup of [22XG[122X. If [22X1 ≠ x ∈ S[122X, then [22XS =
  ⟨ P, x^g ⟩[122X for some [22Xg ∈ S[122X.[133X
  
  [33X[0;0YFirst we show [BG, Proposition 2.2]. Let [22XS[122X be a sporadic simple group, fix a
  Sylow  [22X2[122X-subgroup  [22XP[122X  of  [22XS[122X, and let [22Xx[122X be a nonidentity element in [22XS[122X. We use
  known  information  about  maximal  subgroups of [22XS[122X to show that [22Xx^S[122X is not a
  subset of the union of those maximal subgroups in [22XS[122X that contain [22XP[122X.[133X
  
  [33X[0;0YLet  [22XM[122X  be  a  maximal  subgroup of [22XS[122X with the property [22XP ≤ M[122X. The number of
  [22XS[122X-conjugates of [22XM[122X that contain [22XP[122X is equal to [22X|N_S(P)|/|N_M(P)| ≤ [N_S(P):P][122X,
  thus  these subgroups can contain at most [22X[N_S(P):P] |x^S ∩ M|[122X elements from
  the class [22Xx^S[122X.[133X
  
  [33X[0;0YThus  the  number  of  elements  in [22Xx^S[122X that generate a proper subgroup of [22XS[122X
  together with [22XP[122X is bounded from above by [22X[N_S(P):P] ∑_M |x^S ∩ M|[122X, where the
  sum  is  taken  over  representatives  [22XM[122X  of  conjugacy  classes  of maximal
  subgroups of odd index in [22XS[122X.[133X
  
  [33X[0;0YLet  [22X1_M^S[122X denote the permutation character of the action of [22XS[122X on the cosets
  of  [22XM[122X. We have [22X|x^S ∩ M| = |x^S| 1_M^S(x) / 1_M^S(1)[122X. Hence we are done when
  we show that[133X
  
  
  [24X[33X[0;6Y[N_S(P):P] ∑_M 1_M^S(x) / 1_M^S(1) < 1[133X[124X
  
  [33X[0;0Yholds.[133X
  
  [33X[0;0YThe  numbers  [22X[N_S(P):P][122X  can be read off from [Wil98, Table I]. Here we use
  the  fact  that  the  character  tables  of  the  Sylow [22X2[122X-normalizer of [22XS[122X is
  available except if [22XS[122X is one of [22XCo_1[122X, [22XJ_4[122X, [22XF_3+[122X, [22XB[122X, or [22XM[122X, and that the Sylow
  [22X2[122X-subgroup if self-normalizing in these cases.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xnames:= AllCharacterTableNames( IsSporadicSimple, true,[127X[104X
    [4X[25X>[125X [27X           IsDuplicateTable, false : OrderedBy:= Size );[127X[104X
    [4X[28X[ "M11", "M12", "J1", "M22", "J2", "M23", "HS", "J3", "M24", "McL", [128X[104X
    [4X[28X  "He", "Ru", "Suz", "ON", "Co3", "Co2", "Fi22", "HN", "Ly", "Th", [128X[104X
    [4X[28X  "Fi23", "Co1", "J4", "F3+", "B", "M" ][128X[104X
    [4X[25Xgap>[125X [27Xnormindices:= rec( Co1:= 1, J4:= 1, F3\+:= 1, B:= 1, M:= 1 );;[127X[104X
    [4X[25Xgap>[125X [27Xfor name in names do[127X[104X
    [4X[25X>[125X [27X     n:= CharacterTable( Concatenation( name, "N2" ) );[127X[104X
    [4X[25X>[125X [27X     if n = fail then[127X[104X
    [4X[25X>[125X [27X       Print( name, "\n" );[127X[104X
    [4X[25X>[125X [27X     else[127X[104X
    [4X[25X>[125X [27X       2part:= 2^Length( Positions( Factors( Size( n ) ), 2 ) );[127X[104X
    [4X[25X>[125X [27X       normindices.( name ):= Size( n ) / 2part;[127X[104X
    [4X[25X>[125X [27X     fi;[127X[104X
    [4X[25X>[125X [27X   od;[127X[104X
    [4X[28XCo1[128X[104X
    [4X[28XJ4[128X[104X
    [4X[28XF3+[128X[104X
    [4X[28XB[128X[104X
    [4X[28XM[128X[104X
  [4X[32X[104X
  
  [33X[0;0YFor  all  sporadic  simple  groups [22XS[122X except the Monster group, the primitive
  permutation  characters  [22X1_M^S[122X  can  be computed from the data about maximal
  subgroups contained in [5XGAP[105X's library of character tables.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xmaxbound:= [];;[127X[104X
    [4X[25Xgap>[125X [27Xfor name in Filtered( names, x -> x <> "M" ) do[127X[104X
    [4X[25X>[125X [27X     t:= CharacterTable( name );[127X[104X
    [4X[25X>[125X [27X     mx:= List( Maxes( t ), CharacterTable );[127X[104X
    [4X[25X>[125X [27X     odd:= Filtered( mx, s -> ( Size( t ) / Size( s ) ) mod 2 <> 0 );[127X[104X
    [4X[25X>[125X [27X     primperm:= List( odd, s -> TrivialCharacter( s )^t );[127X[104X
    [4X[25X>[125X [27X     sum:= normindices.( name ) * Sum( primperm, pi -> pi / pi[1] );[127X[104X
    [4X[25X>[125X [27X     Add( maxbound,[127X[104X
    [4X[25X>[125X [27X          [ name, Maximum( sum{ [ 2 .. Length( sum ) ] } ) ] );[127X[104X
    [4X[25X>[125X [27X   od;[127X[104X
    [4X[25Xgap>[125X [27XSortBy( maxbound, x -> - x[2] );[127X[104X
    [4X[25Xgap>[125X [27Xmaxbound[1];[127X[104X
    [4X[28X[ "J2", 3/5 ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YWe  see  that the left hand side of the above inequality is always less than
  or equal to [22X3/5[122X, in particular it is less than [22X1[122X.[133X
  
  [33X[0;0YThe  Monster  group  is  known  to  contain  exactly five classes of maximal
  subgroups  of  odd index, of the structures [22X2^1+24.Co_1[122X (the normalizer of a
  [10X2B[110X  element  in  the  Monster),  [22X2^10+16.O_10^+(2)[122X,  [22X2^2+11+22.(M_24 × S_3)[122X,
  [22X2^5+10+20.(S_3   ×   L_5(2))[122X,  [22X[2^39].(L_3(2)  ×  3S_6)[122X.  The  corresponding
  permutation  characters  are  known,  see  Section  [14X8.16[114X.  First we read the
  information  about the known primitive permutation characters of the Monster
  into  the  [5XGAP[105X  session, and extract the primitive permutation characters of
  odd degree.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xdir:= DirectoriesPackageLibrary( "ctbllib", "data" );;[127X[104X
    [4X[25Xgap>[125X [27Xfilename:= Filename( dir, "prim_perm_M.json" );;[127X[104X
    [4X[25Xgap>[125X [27XMonster_prim_data:= EvalString( StringFile( filename ) )[2];;[127X[104X
    [4X[25Xgap>[125X [27XLength( Monster_prim_data );[127X[104X
    [4X[28X44[128X[104X
    [4X[25Xgap>[125X [27Xt:= CharacterTable( "M" );;[127X[104X
    [4X[25Xgap>[125X [27Xmonstermaxindices:= [];;[127X[104X
    [4X[25Xgap>[125X [27Xmonstermaxtables:= [];;[127X[104X
    [4X[25Xgap>[125X [27Xfor entry in Monster_prim_data do[127X[104X
    [4X[25X>[125X [27X     if Length( entry ) = 1 then[127X[104X
    [4X[25X>[125X [27X       s:= CharacterTable( entry[1] );[127X[104X
    [4X[25X>[125X [27X       Add( monstermaxtables, s );[127X[104X
    [4X[25X>[125X [27X       Add( monstermaxindices, Size( t ) / Size( s ) );[127X[104X
    [4X[25X>[125X [27X     else[127X[104X
    [4X[25X>[125X [27X       Add( monstermaxtables, fail );[127X[104X
    [4X[25X>[125X [27X       Add( monstermaxindices, entry[2][1] );[127X[104X
    [4X[25X>[125X [27X     fi;[127X[104X
    [4X[25X>[125X [27X   od;[127X[104X
    [4X[25Xgap>[125X [27Xodd_prim:= [];;[127X[104X
    [4X[25Xgap>[125X [27Xfor i in [ 1 .. Length( Monster_prim_data ) ] do[127X[104X
    [4X[25X>[125X [27X     if monstermaxindices[i] mod 2 <> 0 then[127X[104X
    [4X[25X>[125X [27X       if monstermaxtables[i] <> fail then[127X[104X
    [4X[25X>[125X [27X         Add( odd_prim, TrivialCharacter( monstermaxtables[i] )^t );[127X[104X
    [4X[25X>[125X [27X       else[127X[104X
    [4X[25X>[125X [27X         Add( odd_prim, Monster_prim_data[i][2] );[127X[104X
    [4X[25X>[125X [27X       fi;[127X[104X
    [4X[25X>[125X [27X     fi;[127X[104X
    [4X[25X>[125X [27X   od;[127X[104X
    [4X[25Xgap>[125X [27XLength( odd_prim );[127X[104X
    [4X[28X5[128X[104X
  [4X[32X[104X
  
  [33X[0;0YNow we can use the same approach as before.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xsum:= normindices.M * Sum( odd_prim, pi -> pi / pi[1] );;[127X[104X
    [4X[25Xgap>[125X [27Xmax:= Maximum( sum{ [ 2 .. Length( sum ) ] } );[127X[104X
    [4X[28X12784979/103007903752128375[128X[104X
    [4X[25Xgap>[125X [27Xmax < 10^-9;[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [33X[0;0YNext we show [BG, Theorem 7.1] and its stronger version stated above. Let us
  first assume that [22XS[122X is not the Monster.[133X
  
  [33X[0;0YAs  a  first step, we generalize the approach from the above computations in
  order  to  check  for which prime divisors [22Xp[122X of [22X|S|[122X and for which nontrivial
  conjugacy  classes [22Xx^S[122X of [22XS[122X the group [22XS[122X is generated by a Sylow [22Xp[122X-subgroup [22XP[122X
  together with a conjugate of [22Xx[122X.[133X
  
  [33X[0;0YThe  upper  bound [22X[N_S(P):P][122X for [22X|N_S(P)|/|N_M(P)|[122X, for a maximal subgroup [22XM[122X
  of  [22XS[122X  that  contains  [22XP[122X, is not good enough in some of the cases considered
  here.  Instead of it, we compute the upper bound [22Xu(S, M, p)[122X which is defined
  as follows; we assume that we know [22X|N_S(P)|[122X.[133X
  
  [30X    [33X[0;6YIf  [22XP[122X  is cyclic then we can compute [22X|N_M(P)|[122X from the character table
        of [22XM[122X, and set [22Xu(S, M, p) = |N_S(P)| / |N_M(P)|[122X.[133X
  
  [30X    [33X[0;6YOtherwise, if [22XP[122X is normal in [22XM[122X, we set [22Xu(S, M, p) = |N_S(P)| / |M|[122X.[133X
  
  [30X    [33X[0;6YOtherwise, if we know a subgroup [22XU[122X of [22XM[122X such that [22XP[122X is a proper normal
        subgroup of [22XU[122X, we set [22Xu(S, M, p) = |N_S(P)| / |U|[122X.[133X
  
  [30X    [33X[0;6YOtherwise, we set [22Xu(S, M, p) = |N_S(P)| / |P|[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xupper_bound:= function( tblS, tblM, p )[127X[104X
    [4X[25X>[125X [27X   local ppart, ppartposS, ppartposM, n, N_S, f, subname, u;[127X[104X
    [4X[25X>[125X [27X[127X[104X
    [4X[25X>[125X [27X   ppart:= Product( Filtered( Factors( Size( tblS ) ), x -> x = p ), 1 );[127X[104X
    [4X[25X>[125X [27X   ppartposS:= Positions( OrdersClassRepresentatives( tblS ), ppart );[127X[104X
    [4X[25X>[125X [27X   if 0 < Length( ppartposS ) then[127X[104X
    [4X[25X>[125X [27X     # P is cyclic.[127X[104X
    [4X[25X>[125X [27X     if tblM = fail then[127X[104X
    [4X[25X>[125X [27X       return ( SizesCentralizers( tblS )[ ppartposS[1] ] * Phi( ppart )[127X[104X
    [4X[25X>[125X [27X                / Length( ppartposS ) ) / ppart;[127X[104X
    [4X[25X>[125X [27X     else[127X[104X
    [4X[25X>[125X [27X       ppartposM:= Positions( OrdersClassRepresentatives( tblM ), ppart );[127X[104X
    [4X[25X>[125X [27X       return ( SizesCentralizers( tblS )[ ppartposS[1] ] * Phi( ppart )[127X[104X
    [4X[25X>[125X [27X                / Length( ppartposS ) ) /[127X[104X
    [4X[25X>[125X [27X              ( SizesCentralizers( tblM )[ ppartposM[1] ] * Phi( ppart )[127X[104X
    [4X[25X>[125X [27X                / Length( ppartposM ) );[127X[104X
    [4X[25X>[125X [27X     fi;[127X[104X
    [4X[25X>[125X [27X   fi;[127X[104X
    [4X[25X>[125X [27X [127X[104X
    [4X[25X>[125X [27X   # Compute |N_S(P)|.[127X[104X
    [4X[25X>[125X [27X   n:= CharacterTable( Concatenation( Identifier( tblS ), "N",[127X[104X
    [4X[25X>[125X [27X                           String( p ) ) );[127X[104X
    [4X[25X>[125X [27X   if n <> fail then[127X[104X
    [4X[25X>[125X [27X     N_S:= Size( n );[127X[104X
    [4X[25X>[125X [27X   elif p = 2 then[127X[104X
    [4X[25X>[125X [27X     N_S:= ppart * normindices.( Identifier( tblS ) );[127X[104X
    [4X[25X>[125X [27X   elif Identifier( tblS ) = "M" and p = 3 then[127X[104X
    [4X[25X>[125X [27X     # The Sylow 3-normalizer is contained in 3^(3+2+6+6):(L3(3)xSD16)[127X[104X
    [4X[25X>[125X [27X     N_S:= ppart * 2^6;[127X[104X
    [4X[25X>[125X [27X   elif Identifier( tblS ) = "F3+" and p = 3 then[127X[104X
    [4X[25X>[125X [27X     N_S:= ppart * 8;[127X[104X
    [4X[25X>[125X [27X   else[127X[104X
    [4X[25X>[125X [27X     Error( "cannot compute |N_S(P)|" );[127X[104X
    [4X[25X>[125X [27X   fi;[127X[104X
    [4X[25X>[125X [27X [127X[104X
    [4X[25X>[125X [27X   if tblM = fail then[127X[104X
    [4X[25X>[125X [27X     return N_S / ppart;[127X[104X
    [4X[25X>[125X [27X   elif Sum( SizesConjugacyClasses( tblM ){[127X[104X
    [4X[25X>[125X [27X                 ClassPositionsOfPCore( tblM, p ) } ) = ppart then[127X[104X
    [4X[25X>[125X [27X     # P is normal in M.[127X[104X
    [4X[25X>[125X [27X     return N_S / Size( tblM );[127X[104X
    [4X[25X>[125X [27X   fi;[127X[104X
    [4X[25X>[125X [27X[127X[104X
    [4X[25X>[125X [27X   # Inspect known character tables of subgroups of M.[127X[104X
    [4X[25X>[125X [27X   f:= N_S / ppart;[127X[104X
    [4X[25X>[125X [27X   for subname in NamesOfFusionSources( tblM ) do[127X[104X
    [4X[25X>[125X [27X     u:= CharacterTable( subname );[127X[104X
    [4X[25X>[125X [27X     if ClassPositionsOfKernel( GetFusionMap( u, tblM ) ) = [ 1 ] and[127X[104X
    [4X[25X>[125X [27X        Sum( SizesConjugacyClasses( u ){[127X[104X
    [4X[25X>[125X [27X                 ClassPositionsOfPCore( u, p ) } ) = ppart then[127X[104X
    [4X[25X>[125X [27X       f:= Minimum( f, N_S / Size( u ) );[127X[104X
    [4X[25X>[125X [27X     fi;[127X[104X
    [4X[25X>[125X [27X   od;[127X[104X
    [4X[25X>[125X [27X[127X[104X
    [4X[25X>[125X [27X   return f;[127X[104X
    [4X[25X>[125X [27X end;;[127X[104X
  [4X[32X[104X
  
  [33X[0;0YWe  run over the sporadic simple groups (except the Monster), and collect in
  the  list  [10Xbadcases_strong[110X  those  [21Xbad[121X prime divisors [22Xp[122X of [22X|S|[122X and conjugacy
  class representatives [22Xx[122X of nonidentity elements in [22XS[122X for which[133X
  
  
  [24X[33X[0;6Y∑_M u(S, M, p) 1_M^S(x) / 1_M^S(1) ≥ 1[133X[124X
  
  [33X[0;0Yholds, where the sum is taken over representatives [22XM[122X of conjugacy classes of
  maximal  subgroups of [22XS[122X whose index in [22XS[122X is coprime to [22Xp[122X. In these cases, we
  have to find other arguments.[133X
  
  [33X[0;0YFor  the  proof  of [BG, Theorem 7.1], we can discard all those entries from
  the  list  of  [21Xbad[121X  [22Xp[122X and [22Xx[122X where [22Xx[122X is not a [22Xq[122X-element, for some prime [22Xq[122X, or
  where  another nonidentity [22Xq[122X-element exists that does not occur in the list.
  This  is  done by collecting a second list [10Xbadcases_thm[110X of the remaining [21Xbad[121X
  cases.[133X
  
  [33X[0;0YFor  the  proof  of  the  stronger version, we will later explicitly compute
  group  elements from the classes in question that generate [22XS[122X together with a
  Sylow  [22Xp[122X-subgroup. (The only technical complication is that the class fusion
  of  maximal  subgroups  of  the type [22X(2^2 × F_4(2)):2[122X of the Baby Monster is
  currently  not  known, thus we cannot simply induce the trivial character in
  this  case. However, the permutation character is uniquely determined by the
  two character tables.)[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xbadcases_thm:= [];;[127X[104X
    [4X[25Xgap>[125X [27Xbadcases_strong:= [];;[127X[104X
    [4X[25Xgap>[125X [27Xfor name in Filtered( names, x -> x <> "M" ) do[127X[104X
    [4X[25X>[125X [27X     t:= CharacterTable( name );[127X[104X
    [4X[25X>[125X [27X     orders:= OrdersClassRepresentatives( t );[127X[104X
    [4X[25X>[125X [27X     n:= NrConjugacyClasses( t );[127X[104X
    [4X[25X>[125X [27X     mx:= List( Maxes( t ), CharacterTable );[127X[104X
    [4X[25X>[125X [27X     for p in PrimeDivisors( Size( t ) ) do[127X[104X
    [4X[25X>[125X [27X       good:= Filtered( mx, s -> ( Size( t ) / Size( s ) ) mod p <> 0 );[127X[104X
    [4X[25X>[125X [27X       primperm:= [];[127X[104X
    [4X[25X>[125X [27X       for s in good do[127X[104X
    [4X[25X>[125X [27X         if GetFusionMap( s, t ) <> fail then[127X[104X
    [4X[25X>[125X [27X           Add( primperm, TrivialCharacter( s )^t );[127X[104X
    [4X[25X>[125X [27X         else[127X[104X
    [4X[25X>[125X [27X           ind:= Set( PossibleClassFusions( s, t ),[127X[104X
    [4X[25X>[125X [27X                      map -> InducedClassFunctionsByFusionMap( s, t,[127X[104X
    [4X[25X>[125X [27X                                 [ TrivialCharacter( s ) ], map )[1] );[127X[104X
    [4X[25X>[125X [27X           if Length( ind ) <> 1 then[127X[104X
    [4X[25X>[125X [27X             Error( "permutation character not uniquely determined" );[127X[104X
    [4X[25X>[125X [27X           fi;[127X[104X
    [4X[25X>[125X [27X           Add( primperm, ind[1] );[127X[104X
    [4X[25X>[125X [27X         fi;[127X[104X
    [4X[25X>[125X [27X       od;[127X[104X
    [4X[25X>[125X [27X       sum:= Sum( [ 1 .. Length( good ) ],[127X[104X
    [4X[25X>[125X [27X                  i -> upper_bound( t, good[i], p )[127X[104X
    [4X[25X>[125X [27X                       * primperm[i] / primperm[i][1] );[127X[104X
    [4X[25X>[125X [27X       badpos:= Filtered( [ 2 .. Length( sum ) ], i -> sum[i] >= 1 );[127X[104X
    [4X[25X>[125X [27X       if badpos <> [] then[127X[104X
    [4X[25X>[125X [27X         Add( badcases_strong, [ name, p, ShallowCopy( badpos ) ] );[127X[104X
    [4X[25X>[125X [27X         for i in ShallowCopy( badpos ) do[127X[104X
    [4X[25X>[125X [27X           q:= SmallestRootInt( orders[i] );[127X[104X
    [4X[25X>[125X [27X           if IsPrimeInt( q ) then[127X[104X
    [4X[25X>[125X [27X             if ForAny( [ 2 .. n ],[127X[104X
    [4X[25X>[125X [27X                        j -> SmallestRootInt( orders[j] ) = q[127X[104X
    [4X[25X>[125X [27X                             and not j in badpos ) then[127X[104X
    [4X[25X>[125X [27X               RemoveSet( badpos, i );[127X[104X
    [4X[25X>[125X [27X             fi;[127X[104X
    [4X[25X>[125X [27X           fi;[127X[104X
    [4X[25X>[125X [27X         od;[127X[104X
    [4X[25X>[125X [27X         if not IsEmpty( badpos ) then[127X[104X
    [4X[25X>[125X [27X           Add( badcases_thm, [ name, p, badpos ] );[127X[104X
    [4X[25X>[125X [27X         fi;[127X[104X
    [4X[25X>[125X [27X       fi;[127X[104X
    [4X[25X>[125X [27X     od;[127X[104X
    [4X[25X>[125X [27X   od;[127X[104X
    [4X[25Xgap>[125X [27Xbadcases_thm;[127X[104X
    [4X[28X[ [ "M23", 3, [ 3 ] ], [ "HS", 3, [ 4, 11 ] ] ][128X[104X
    [4X[25Xgap>[125X [27Xbadcases_strong;[127X[104X
    [4X[28X[ [ "M11", 5, [ 2 ] ], [ "M12", 5, [ 3, 4 ] ], [ "M22", 3, [ 2 ] ], [128X[104X
    [4X[28X  [ "M22", 5, [ 2 ] ], [ "J2", 3, [ 2 ] ], [ "M23", 3, [ 2, 3 ] ], [128X[104X
    [4X[28X  [ "M23", 5, [ 2 ] ], [ "M23", 7, [ 2 ] ], [128X[104X
    [4X[28X  [ "HS", 3, [ 2, 3, 4, 5, 6, 7, 9, 11 ] ], [ "HS", 5, [ 2, 3, 5 ] ], [128X[104X
    [4X[28X  [ "M24", 5, [ 2, 4 ] ], [ "M24", 7, [ 2, 4 ] ], [ "He", 5, [ 2 ] ], [128X[104X
    [4X[28X  [ "Co2", 3, [ 2, 3 ] ], [ "Fi22", 5, [ 2 ] ], [ "Fi22", 7, [ 2 ] ], [128X[104X
    [4X[28X  [ "Fi23", 5, [ 2, 3, 5 ] ], [ "Fi23", 7, [ 2 ] ], [ "B", 7, [ 2 ] ][128X[104X
    [4X[28X ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YMost  of these open cases can be ruled out by constructing the group [22XS[122X and a
  Sylow  [22Xp[122X-subgroup  [22XP[122X  in  question and then finding explicit elements [22Xx[122X such
  that  [22XS[122X is generated by [22XP[122X and [22Xx[122X. For that, we use the data from the [5XAtlas[105X of
  Group Representations [WWT+].[133X
  
  [33X[0;0YThe  following  function  tries  to  find random elements from all conjugacy
  classes  of  nonidentity elements that have the desired property. It returns
  [9Xfail[109X   if  no  straight  line  program  is  available  for  computing  class
  representatives,  and  returns  [22XP[122X and the list of class representatives that
  generate together with [22XP[122X if such elements were found. Thus the function will
  not return if the generation property does not hold.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xprove_generation:= function( name, p )[127X[104X
    [4X[25X>[125X [27X   local S, prg, P, reps, good, x, g, U;[127X[104X
    [4X[25X>[125X [27X[127X[104X
    [4X[25X>[125X [27X   prg:= AtlasProgram( name, "classes" );[127X[104X
    [4X[25X>[125X [27X   if prg = fail then[127X[104X
    [4X[25X>[125X [27X     return fail;[127X[104X
    [4X[25X>[125X [27X   fi;[127X[104X
    [4X[25X>[125X [27X[127X[104X
    [4X[25X>[125X [27X   S:= AtlasGroup( name );[127X[104X
    [4X[25X>[125X [27X   P:= SylowSubgroup( S, p );[127X[104X
    [4X[25X>[125X [27X   reps:= ResultOfStraightLineProgram( prg.program, GeneratorsOfGroup( S ) );[127X[104X
    [4X[25X>[125X [27X   good:= [];[127X[104X
    [4X[25X>[125X [27X   for x in Filtered( reps, x -> Order( x ) <> 1 ) do[127X[104X
    [4X[25X>[125X [27X     repeat[127X[104X
    [4X[25X>[125X [27X       g:= Random( S );[127X[104X
    [4X[25X>[125X [27X       U:= ClosureGroup( P, x^g );[127X[104X
    [4X[25X>[125X [27X     until Size( U ) = Size( S );[127X[104X
    [4X[25X>[125X [27X     Add( good, x^g );[127X[104X
    [4X[25X>[125X [27X   od;[127X[104X
    [4X[25X>[125X [27X[127X[104X
    [4X[25X>[125X [27X   return [ P, good ];[127X[104X
    [4X[25X>[125X [27X end;;[127X[104X
    [4X[25Xgap>[125X [27Xfor entry in badcases_strong do[127X[104X
    [4X[25X>[125X [27X     res:= prove_generation( entry[1], entry[2] );[127X[104X
    [4X[25X>[125X [27X     if res = fail then[127X[104X
    [4X[25X>[125X [27X       Print( "no classes script for ", entry, "\n" );[127X[104X
    [4X[25X>[125X [27X     fi;[127X[104X
    [4X[25X>[125X [27X   od;[127X[104X
    [4X[28Xno classes script for [ "He", 5, [ 2 ] ][128X[104X
    [4X[28Xno classes script for [ "Fi22", 5, [ 2 ] ][128X[104X
    [4X[28Xno classes script for [ "Fi22", 7, [ 2 ] ][128X[104X
    [4X[28Xno classes script for [ "Fi23", 5, [ 2, 3, 5 ] ][128X[104X
    [4X[28Xno classes script for [ "Fi23", 7, [ 2 ] ][128X[104X
    [4X[28Xno classes script for [ "B", 7, [ 2 ] ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YIn  the  remaining  six  cases, we show only the generation property for the
  class  representatives in the list. These are involutions from the class [10X2A[110X,
  and  for the group [22XFi_23[122X and [22Xp = 5[122X additionally elements from the classes [10X2B[110X
  and [10X3A[110X.[133X
  
  [33X[0;0YA  [10X2A[110X element in the group [22XHe[122X can be found as the fifth power of any element
  of order [22X10[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XS:= AtlasGroup( "He" );;[127X[104X
    [4X[25Xgap>[125X [27Xrepeat[127X[104X
    [4X[25X>[125X [27X     x:= Random( S );[127X[104X
    [4X[25X>[125X [27X   until Order( x ) = 10;[127X[104X
    [4X[25Xgap>[125X [27Xx:= x^5;;[127X[104X
    [4X[25Xgap>[125X [27XP5:= SylowSubgroup( S, 5 );;[127X[104X
    [4X[25Xgap>[125X [27Xrepeat[127X[104X
    [4X[25X>[125X [27X     g:= Random( S );[127X[104X
    [4X[25X>[125X [27X     U:= ClosureGroup( P5, x^g );[127X[104X
    [4X[25X>[125X [27X   until Size( U ) = Size( S );[127X[104X
  [4X[32X[104X
  
  [33X[0;0YA  [10X2A[110X  element  in  the  group  [22XFi_22[122X can be found as the [22X15[122X-th power of any
  element of order [22X30[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XS:= AtlasGroup( "Fi22" );;[127X[104X
    [4X[25Xgap>[125X [27Xrepeat[127X[104X
    [4X[25X>[125X [27X     x:= Random( S );[127X[104X
    [4X[25X>[125X [27X   until Order( x ) = 30;[127X[104X
    [4X[25Xgap>[125X [27Xx:= x^15;;[127X[104X
    [4X[25Xgap>[125X [27XP5:= SylowSubgroup( S, 5 );;[127X[104X
    [4X[25Xgap>[125X [27Xrepeat[127X[104X
    [4X[25X>[125X [27X     g:= Random( S );[127X[104X
    [4X[25X>[125X [27X     U:= ClosureGroup( P5, x^g );[127X[104X
    [4X[25X>[125X [27X   until Size( U ) = Size( S );[127X[104X
    [4X[25Xgap>[125X [27XP7:= SylowSubgroup( S, 7 );;[127X[104X
    [4X[25Xgap>[125X [27Xrepeat[127X[104X
    [4X[25X>[125X [27X     g:= Random( S );[127X[104X
    [4X[25X>[125X [27X     U:= ClosureGroup( P7, x^g );[127X[104X
    [4X[25X>[125X [27X   until Size( U ) = Size( S );[127X[104X
  [4X[32X[104X
  
  [33X[0;0YA  [10X2A[110X  element  in  the  group  [22XFi_23[122X can be found as the [22X21[122X-st power of any
  element of order [22X42[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XS:= AtlasGroup( "Fi23" );;[127X[104X
    [4X[25Xgap>[125X [27Xrepeat[127X[104X
    [4X[25X>[125X [27X     x:= Random( S );[127X[104X
    [4X[25X>[125X [27X   until Order( x ) = 42;[127X[104X
    [4X[25Xgap>[125X [27Xx:= x^21;;[127X[104X
    [4X[25Xgap>[125X [27XP5:= SylowSubgroup( S, 5 );;[127X[104X
    [4X[25Xgap>[125X [27Xrepeat[127X[104X
    [4X[25X>[125X [27X     g:= Random( S );[127X[104X
    [4X[25X>[125X [27X     U:= ClosureGroup( P5, x^g );[127X[104X
    [4X[25X>[125X [27X   until Size( U ) = Size( S );[127X[104X
    [4X[25Xgap>[125X [27XP7:= SylowSubgroup( S, 7 );;[127X[104X
    [4X[25Xgap>[125X [27Xrepeat[127X[104X
    [4X[25X>[125X [27X     g:= Random( S );[127X[104X
    [4X[25X>[125X [27X     U:= ClosureGroup( P7, x^g );[127X[104X
    [4X[25X>[125X [27X   until Size( U ) = Size( S );[127X[104X
  [4X[32X[104X
  
  [33X[0;0YA  [10X2B[110X  element  in  the  group  [22XFi_23[122X can be found as the [22X30[122X-th power of any
  element of order [22X60[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xrepeat[127X[104X
    [4X[25X>[125X [27X     x:= Random( S );[127X[104X
    [4X[25X>[125X [27X   until Order( x ) = 60;[127X[104X
    [4X[25Xgap>[125X [27Xx:= x^30;;[127X[104X
    [4X[25Xgap>[125X [27Xrepeat[127X[104X
    [4X[25X>[125X [27X     g:= Random( S );[127X[104X
    [4X[25X>[125X [27X     U:= ClosureGroup( P5, x^g );[127X[104X
    [4X[25X>[125X [27X   until Size( U ) = Size( S );[127X[104X
  [4X[32X[104X
  
  [33X[0;0YA  [10X3A[110X  element  in  the  group  [22XFi_23[122X can be found as the [22X20[122X-th power of any
  element of order [22X60[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xrepeat[127X[104X
    [4X[25X>[125X [27X     x:= Random( S );[127X[104X
    [4X[25X>[125X [27X   until Order( x ) = 60;[127X[104X
    [4X[25Xgap>[125X [27Xx:= x^20;;[127X[104X
    [4X[25Xgap>[125X [27Xrepeat[127X[104X
    [4X[25X>[125X [27X     g:= Random( S );[127X[104X
    [4X[25X>[125X [27X     U:= ClosureGroup( P5, x^g );[127X[104X
    [4X[25X>[125X [27X   until Size( U ) = Size( S );[127X[104X
  [4X[32X[104X
  
  [33X[0;0YIn  the  open  case  for the Baby Monster, we have to show that the group is
  generated  by  a  [10X2A[110X  element  and  an  element of order [22X7[122X. This can be done
  character-theoretically,  for  example as follows. There are such elements [22Xx[122X
  and  [22Xy[122X  whose product [22Xx y[122X has order [22X47[122X, and the only proper subgroups of the
  Baby  Monster  that  contain  elements  of order [22X47[122X are contained in maximal
  subgroups of the type [22X47:23[122X. Thus [22Xx[122X and [22Xy[122X generate the Baby Monster.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xt:= CharacterTable( "B" );;[127X[104X
    [4X[25Xgap>[125X [27X7pos:= Positions( OrdersClassRepresentatives( t ), 7 );[127X[104X
    [4X[28X[ 31 ][128X[104X
    [4X[25Xgap>[125X [27X47pos:= Positions( OrdersClassRepresentatives( t ), 47 );[127X[104X
    [4X[28X[ 172, 173 ][128X[104X
    [4X[25Xgap>[125X [27XClassMultiplicationCoefficient( t, 2, 7pos[1], 47pos[1] );[127X[104X
    [4X[28X7332[128X[104X
    [4X[25Xgap>[125X [27XFiltered( Maxes( t ),[127X[104X
    [4X[25X>[125X [27X       x -> Size( CharacterTable( x ) ) mod 47 = 0 );[127X[104X
    [4X[28X[ "47:23" ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YNow  consider  the  case that [22XS[122X is the Monster, which is special because the
  complete  list  of classes of maximal subgroups of [22XS[122X is currently not known.
  From [NW13] and [Wil] we know [22X44[122X classes of maximal subgroups, and that each
  possible additional maximal subgroup is almost simple and has socle [22XL_2(13)[122X,
  [22XU_3(4)[122X,  [22XU_3(8)[122X,  or  [22XSz(8)[122X.  This  implies  that  we know all those maximal
  subgroups  that  contain  a Sylow-[22Xp[122X-subgroup of [22XS[122X except in the case [22Xp = 19[122X,
  where maximal subgroups with socle [22XU_3(8)[122X may arise.[133X
  
  [33X[0;0YThus  let  us first consider that at least one of [22Xp[122X, [22Xr[122X is different from [22X19[122X.
  In this situation, we use the same approach as for the other sporadic simple
  groups.  The only complication is that not all permutation characters [22X1_M^S[122X,
  for  the  relevant  maximal  subgroups  [22XM[122X  of [22XS[122X, are known; however, if this
  happens  then  the  character  table  of  [22XM[122X is known, and we can compute the
  possible  permutation characters, and take the common upper bounds for these
  characters. In each case, we get that the claimed property holds.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xt:= CharacterTable( "M" );;[127X[104X
    [4X[25Xgap>[125X [27Xorders:= OrdersClassRepresentatives( t );;[127X[104X
    [4X[25Xgap>[125X [27Xfor p in Difference( PrimeDivisors( Size( t ) ), [ 19 ] ) do[127X[104X
    [4X[25X>[125X [27X  goodpos:= Filtered( [ 1 .. Length( Monster_prim_data ) ],[127X[104X
    [4X[25X>[125X [27X                      i -> monstermaxindices[i] mod p <> 0 );[127X[104X
    [4X[25X>[125X [27X  sum:= ListWithIdenticalEntries( NrConjugacyClasses( t ), 0 );[127X[104X
    [4X[25X>[125X [27X  for i in goodpos do[127X[104X
    [4X[25X>[125X [27X    if Length( Monster_prim_data[i] ) = 2 then[127X[104X
    [4X[25X>[125X [27X      # We know the permutation character but not the subgroup table.[127X[104X
    [4X[25X>[125X [27X      sum:= sum + upper_bound( t, fail, p )[127X[104X
    [4X[25X>[125X [27X                  * Monster_prim_data[i][2] / monstermaxindices[i];[127X[104X
    [4X[25X>[125X [27X    else[127X[104X
    [4X[25X>[125X [27X      s:= monstermaxtables[i];[127X[104X
    [4X[25X>[125X [27X      if GetFusionMap( s, t ) <> fail then[127X[104X
    [4X[25X>[125X [27X        # We can compute the permutation character.[127X[104X
    [4X[25X>[125X [27X        sum:= sum + upper_bound( t, s, p )[127X[104X
    [4X[25X>[125X [27X                    * TrivialCharacter( s )^t / monstermaxindices[i];[127X[104X
    [4X[25X>[125X [27X      else[127X[104X
    [4X[25X>[125X [27X        # We get only candidates for the permutation character.[127X[104X
    [4X[25X>[125X [27X        cand:= Set( PossibleClassFusions( s, t ),[127X[104X
    [4X[25X>[125X [27X                    map -> InducedClassFunctionsByFusionMap( s, t,[127X[104X
    [4X[25X>[125X [27X                               [ TrivialCharacter( s ) ], map )[1] );[127X[104X
    [4X[25X>[125X [27X        # For each class, take the maximum of the possible values.[127X[104X
    [4X[25X>[125X [27X        sum:= sum + upper_bound( t, s, p )[127X[104X
    [4X[25X>[125X [27X                    * List( TransposedMat( cand ), Maximum )[127X[104X
    [4X[25X>[125X [27X                    / monstermaxindices[i];[127X[104X
    [4X[25X>[125X [27X      fi;[127X[104X
    [4X[25X>[125X [27X    fi;[127X[104X
    [4X[25X>[125X [27X  od;[127X[104X
    [4X[25X>[125X [27X  badpos:= Filtered( [ 2 .. Length( sum ) ], i -> sum[i] >= 1 );[127X[104X
    [4X[25X>[125X [27X  if badpos <> [] then[127X[104X
    [4X[25X>[125X [27X    Error( "check open cases in ", badpos, "\n" );[127X[104X
    [4X[25X>[125X [27X  fi;[127X[104X
    [4X[25X>[125X [27Xod;[127X[104X
  [4X[32X[104X
  
  [33X[0;0YFinally,  let  [22Xp  = r = 19[122X. The group [22XS[122X has exactly one class of elements of
  order  [22X19[122X.  Let  [22Xx[122X  be  such  an  element. From the character table of [22XS[122X, we
  compute that there exist conjugates [22Xy[122X of [22Xx[122X such that [22Xx y[122X has order [22X71[122X. Since
  [22X⟨ x, y ⟩ = ⟨ x, x y ⟩[122X holds and no maximal subgroup of [22XS[122X has order divisible
  by [22X19 ⋅ 71[122X, we have [22X⟨ x, y ⟩ = S[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xpos19:= Positions( OrdersClassRepresentatives( t ), 19 );[127X[104X
    [4X[28X[ 63 ][128X[104X
    [4X[25Xgap>[125X [27Xpos71:= Positions( OrdersClassRepresentatives( t ), 71 );[127X[104X
    [4X[28X[ 169, 170 ][128X[104X
    [4X[25Xgap>[125X [27XClassMultiplicationCoefficient( t, pos19[1], pos19[1], pos71[1] );[127X[104X
    [4X[28X621743152370566020417806353602387433415016198936[128X[104X
    [4X[25Xgap>[125X [27XForAny( monstermaxindices,[127X[104X
    [4X[25X>[125X [27X           x -> ( Size( t ) / x ) mod ( 19 * 71 ) = 0 );[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XForAny( [ "L2(13)", "U3(4)", "U3(8)", "Sz(8)" ],[127X[104X
    [4X[25X>[125X [27X           x -> Size( CharacterTable( x ) ) mod 71 = 0 );[127X[104X
    [4X[28Xfalse[128X[104X
  [4X[32X[104X
  
