  
  [1X10 [33X[0;0Y[5XGAP[105X[101X[1X computations needed in the proof of [DNT13, Theorem 6.1 (ii)][133X[101X
  
  [33X[0;0YDate: September 19th, 2011[133X
  
  [33X[0;0Y(This is joint work with Klaus Lux.)[133X
  
  [33X[0;0YThis  is a collection of example computations that are cited in the Appendix
  of [DNT13].  In  each case, the aim is to show that the extension of a given
  finite  simple  group  by  an elementary abelian group of given rank has the
  property  that not all complex irreducible characters of the same degree are
  Galois conjugate.[133X
  
  [33X[0;0YThe  purpose of this writeup is twofold. On the one hand, the details of the
  computations  are documented this way. On the other hand, the [5XGAP[105X code shown
  for  the  examples  can  be used as test input for automatic checking of the
  data  and  the  functions  used.}  For the computations, we need some Brauer
  character  tables  from [JLPW95],  some generating matrices from [WWT+], and
  some    functions   from   the   [5XGAP[105X   system [GAP19]   and   its   packages
  [10XAtlasRep[110X [WPN+19], [10Xcohomolo[110X [Hol08], [10XCTblLib[110X [Bre20], and [10XTomLib[110X [NMP18].[133X
  
  [33X[0;0YFirst we load the necessary [5XGAP[105X packages.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XLoadPackage( "AtlasRep", "1.5", false );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XLoadPackage( "cohomolo", "1.6", false );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XLoadPackage( "CTblLib", "1.2", false );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XLoadPackage( "TomLib", "1.2.1", false );[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  
  [1X10.1 [33X[0;0Y[22XG/N ≅ Sz(8)[122X[101X[1X and [22X|N| = 2^12[122X[101X[1X[133X[101X
  
  [33X[0;0YThe  group  [22XS = Sz(8)[122X has exactly one irreducible [22X12[122X-dimensional module over
  the  field with two elements, up to isomorphism. This module can be obtained
  from  any  of  the  three  absolutely irreducible [22X4[122X-dimensional [22XS[122X-modules in
  characteristic two, by regarding it as a module over the prime field [22XGF(2)[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xp:= 2;;  d:= 12;;[127X[104X
    [4X[25Xgap>[125X [27Xt:= CharacterTable( "Sz(8)" ) mod p;[127X[104X
    [4X[28XBrauerTable( "Sz(8)", 2 )[128X[104X
    [4X[25Xgap>[125X [27Xirr:= Filtered( Irr( t ), x -> x[1] <= d );;[127X[104X
    [4X[25Xgap>[125X [27XDisplay( t, rec( chars:= irr, powermap:= false,[127X[104X
    [4X[25X>[125X [27X                    centralizers:= false ) );[127X[104X
    [4X[28XSz(8)mod2[128X[104X
    [4X[28X[128X[104X
    [4X[28X       1a 5a 7a 7b 7c 13a 13b 13c[128X[104X
    [4X[28X[128X[104X
    [4X[28XY.1     1  1  1  1  1   1   1   1[128X[104X
    [4X[28XY.2     4 -1  A  C  B   D   F   E[128X[104X
    [4X[28XY.3     4 -1  B  A  C   E   D   F[128X[104X
    [4X[28XY.4     4 -1  C  B  A   F   E   D[128X[104X
    [4X[28X[128X[104X
    [4X[28XA = E(7)^2+E(7)^3+E(7)^4+E(7)^5[128X[104X
    [4X[28XB = E(7)+E(7)^2+E(7)^5+E(7)^6[128X[104X
    [4X[28XC = E(7)+E(7)^3+E(7)^4+E(7)^6[128X[104X
    [4X[28XD = E(13)+E(13)^5+E(13)^8+E(13)^12[128X[104X
    [4X[28XE = E(13)^4+E(13)^6+E(13)^7+E(13)^9[128X[104X
    [4X[28XF = E(13)^2+E(13)^3+E(13)^10+E(13)^11[128X[104X
    [4X[25Xgap>[125X [27XList( irr, x -> SizeOfFieldOfDefinition( x, p ) );[127X[104X
    [4X[28X[ 2, 8, 8, 8 ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YFirst  we  construct the [22X12[122X-dimensional irreducible representation of [22XS[122X over
  [22XGF(2)[122X,  using  that  the  [5XAtlas[105X  of  Group  Representations  provides matrix
  generators for [22XS[122X in the [22X4[122X-dimensional representation over [22XGF(8)[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xinfo:= OneAtlasGeneratingSetInfo( "Sz(8)", Dimension, 4,[127X[104X
    [4X[25X>[125X [27X              Characteristic, p );[127X[104X
    [4X[28Xrec( charactername := "4a", constituents := [ 2 ], contents := "core",[128X[104X
    [4X[28X  dim := 4, groupname := "Sz(8)", id := "a", [128X[104X
    [4X[28X  identifier := [ "Sz(8)", [ "Sz8G1-f8r4aB0.m1", "Sz8G1-f8r4aB0.m2" ],[128X[104X
    [4X[28X      1, 8 ], repname := "Sz8G1-f8r4aB0", repnr := 17, [128X[104X
    [4X[28X  ring := GF(2^3), size := 29120, standardization := 1, [128X[104X
    [4X[28X  type := "matff" )[128X[104X
    [4X[25Xgap>[125X [27Xgens_dim4:= AtlasGenerators( info ).generators;;[127X[104X
    [4X[25Xgap>[125X [27Xb:= Basis( GF(8) );; [127X[104X
    [4X[25Xgap>[125X [27Xgens_dim12:= List( gens_dim4, x -> BlownUpMatrix( b, x ) );;[127X[104X
  [4X[32X[104X
  
  [33X[0;0YWe claim that any extension of [22XS[122X with the given module splits.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xs:= AtlasGroup( "Sz(8)", IsPermGroup, true );;[127X[104X
    [4X[25Xgap>[125X [27Xchr:= CHR( s, p, 0, gens_dim12 );;[127X[104X
    [4X[25Xgap>[125X [27XSizeScreen( [ 100 ] );;[127X[104X
    [4X[25Xgap>[125X [27XSecondCohomologyDimension( chr );[127X[104X
    [4X[28X0[128X[104X
    [4X[25Xgap>[125X [27XSizeScreen( [ 72 ] );;[127X[104X
  [4X[32X[104X
  
  [33X[0;0Y(The   function  [10XCHR[110X  takes  as  its  arguments  a  permutation  group,  the
  characteristic  of  the  module, a finitely presented group (or zero), and a
  list of matrices that define the module in the sense that they correspond to
  the  generators  of the given permutation group. Note that this condition is
  satisfied   because   the   generators   provided  by  the  [5XAtlas[105X  of  Group
  Representations  are compatible.) So it is enough to consider the semidirect
  product  [22XG = 2^12:Sz(8)[122X. If we would like then we could represent this group
  as  a  group of [22X13 × 13[122X matrices over [22XGF(2)[122X, as follows. For each element of
  [22XG[122X,  the  submatrix consisting of the first [22X12[122X rows and columns describes the
  part from the complement [22XSz(8)[122X, in its action on the module in question, and
  the  last row describes the part from the elementary abelian normal group [22XN[122X;
  the  last  column  is zero, except for an identity entry in the last row. In
  order  to  write  down generators of this group, it suffices to take the two
  generators of the complement plus one nonidentity element from [22XN[122X. (Note that
  [22XN[122X is irreducible.)[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xmats:= List( [1 .. 3 ], x -> IdentityMat( d+1, GF(p) ) );;[127X[104X
    [4X[25Xgap>[125X [27Xv:= mats[1][ d+1 ];;[127X[104X
    [4X[25Xgap>[125X [27Xmats[1]{ [ 1 .. d ] }{ [ 1 .. d ] }:= gens_dim12[1];;[127X[104X
    [4X[25Xgap>[125X [27Xmats[2]{ [ 1 .. d ] }{ [ 1 .. d ] }:= gens_dim12[2];;[127X[104X
    [4X[25Xgap>[125X [27Xmats[3][ d+1 ][1]:= Z(p)^0;;[127X[104X
    [4X[25Xgap>[125X [27Xgrp:= Group( mats );;[127X[104X
    [4X[25Xgap>[125X [27Xg:= Image( IsomorphismPermGroup( grp ) );;[127X[104X
    [4X[25Xgap>[125X [27XSize( g );[127X[104X
    [4X[28X119275520[128X[104X
    [4X[25Xgap>[125X [27XNrConjugacyClasses( g );[127X[104X
    [4X[28X41[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  [5XGAP[105X Character Table Library contains the ordinary character table of [22XG[122X.
  We  check  this  as  follows. By the above cohomology result, the group [22XG[122X is
  uniquely  determined, up to isomorphism, by the group order and the property
  that  [22XG[122X  has  a  minimal  normal  subgroup [22XN[122X such that [22XG/N[122X is a simple group
  isomorphic with [22XS[122X.[133X
  
  [33X[0;0Y(Since  [22X|G|/|S|[122X  is  a  power  of  two,  [22XN[122X  is  a [22X2[122X-group. By the minimality
  condition,  [22XN[122X  is  elementary  abelian  and the action of [22XS[122X on [22XN[122X affords the
  desired [22XS[122X-module. Note that the isomorphism type of a finite simple group is
  determined by its character table.)[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xiso:= IsomorphismTypeInfoFiniteSimpleGroup( s );[127X[104X
    [4X[28Xrec( name := "2B(2,8) = 2C(2,8) = Sz(8)", parameter := 8, [128X[104X
    [4X[28X  series := "2B", shortname := "Sz(8)" )[128X[104X
    [4X[25Xgap>[125X [27Xnames:= AllCharacterTableNames( Size, 2^12 * Size( s ) );;[127X[104X
    [4X[25Xgap>[125X [27Xcand:= List( names, CharacterTable );;[127X[104X
    [4X[25Xgap>[125X [27Xcand:= Filtered( cand,[127X[104X
    [4X[25X>[125X [27X     t -> ForAny( ClassPositionsOfMinimalNormalSubgroups( t ),[127X[104X
    [4X[25X>[125X [27X            n -> IsomorphismTypeInfoFiniteSimpleGroup( t / n ) = iso ) );[127X[104X
    [4X[28X[ CharacterTable( "2^12:Sz(8)" ) ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YSo  we can easily check that [22XG[122X has eight rational valued irreducibles of the
  degree [22X455[122X (or of the degree [22X3640[122X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xt:= cand[1];;[127X[104X
    [4X[25Xgap>[125X [27Xrationals:= Filtered( Irr( t ), x -> IsSubset( Integers, x ) );;[127X[104X
    [4X[25Xgap>[125X [27XCollected( List( rationals, x -> x[1] ) );[127X[104X
    [4X[28X[ [ 1, 1 ], [ 64, 1 ], [ 91, 1 ], [ 455, 8 ], [ 3640, 8 ] ][128X[104X
  [4X[32X[104X
  
  
  [1X10.2 [33X[0;0Y[22XG/N ≅ M_22[122X[101X[1X and [22X|N| = 2^10[122X[101X[1X[133X[101X
  
  [33X[0;0YThe  group  [22XS = M_22[122X has exactly two irreducible [22X10[122X-dimensional modules over
  the  field  with  two elements, up to isomorphism. These modules are in fact
  absolutely irreducible.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xp:= 2;;  d:= 10;;[127X[104X
    [4X[25Xgap>[125X [27Xt:= CharacterTable( "M22" ) mod p;[127X[104X
    [4X[28XBrauerTable( "M22", 2 )[128X[104X
    [4X[25Xgap>[125X [27Xirr:= Filtered( Irr( t ), x -> x[1] <= d );;[127X[104X
    [4X[25Xgap>[125X [27XDisplay( t, rec( chars:= irr, powermap:= false,[127X[104X
    [4X[25X>[125X [27X                    centralizers:= false ) );[127X[104X
    [4X[28XM22mod2[128X[104X
    [4X[28X[128X[104X
    [4X[28X       1a 3a 5a 7a 7b 11a 11b[128X[104X
    [4X[28X[128X[104X
    [4X[28XY.1     1  1  1  1  1   1   1[128X[104X
    [4X[28XY.2    10  1  .  A /A  -1  -1[128X[104X
    [4X[28XY.3    10  1  . /A  A  -1  -1[128X[104X
    [4X[28X[128X[104X
    [4X[28XA = E(7)+E(7)^2+E(7)^4[128X[104X
    [4X[28X  = (-1+Sqrt(-7))/2 = b7[128X[104X
    [4X[25Xgap>[125X [27XList( irr, x -> SizeOfFieldOfDefinition( x, p ) );[127X[104X
    [4X[28X[ 2, 2, 2 ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YFirst  we  construct the two irreducible [22X10[122X-dimensional representations of [22XS[122X
  over [22XGF(2)[122X, again using that the [5XAtlas[105X of Group Representations provides the
  matrix generators in question.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xinfo:= AllAtlasGeneratingSetInfos( "M22", Dimension, d,[127X[104X
    [4X[25X>[125X [27X              Characteristic, p );[127X[104X
    [4X[28X[ rec( charactername := "10a", constituents := [ 2 ], [128X[104X
    [4X[28X      contents := "core", dim := 10, groupname := "M22", id := "a", [128X[104X
    [4X[28X      identifier := [128X[104X
    [4X[28X        [ "M22", [ "M22G1-f2r10aB0.m1", "M22G1-f2r10aB0.m2" ], 1, 2 ],[128X[104X
    [4X[28X      repname := "M22G1-f2r10aB0", repnr := 13, ring := GF(2), [128X[104X
    [4X[28X      size := 443520, standardization := 1, type := "matff" ), [128X[104X
    [4X[28X  rec( charactername := "10b", constituents := [ 3 ], [128X[104X
    [4X[28X      contents := "core", dim := 10, groupname := "M22", id := "b", [128X[104X
    [4X[28X      identifier := [128X[104X
    [4X[28X        [ "M22", [ "M22G1-f2r10bB0.m1", "M22G1-f2r10bB0.m2" ], 1, 2 ],[128X[104X
    [4X[28X      repname := "M22G1-f2r10bB0", repnr := 14, ring := GF(2), [128X[104X
    [4X[28X      size := 443520, standardization := 1, type := "matff" ) ][128X[104X
    [4X[25Xgap>[125X [27Xgens:= List( info, r -> AtlasGenerators( r ).generators );;[127X[104X
  [4X[32X[104X
  
  [33X[0;0YWe claim that any extension of [22XS[122X with any of the two given modules splits.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xs:= AtlasGroup( "M22", IsPermGroup, true );;[127X[104X
    [4X[25Xgap>[125X [27Xchr:= CHR( s, p, 0, gens[1] );;[127X[104X
    [4X[25Xgap>[125X [27XSizeScreen( [ 100 ] );;[127X[104X
    [4X[25Xgap>[125X [27XSecondCohomologyDimension( chr );[127X[104X
    [4X[28X0[128X[104X
    [4X[25Xgap>[125X [27Xchr:= CHR( s, p, 0, gens[2] );;[127X[104X
    [4X[25Xgap>[125X [27XSecondCohomologyDimension( chr );[127X[104X
    [4X[28X0[128X[104X
    [4X[25Xgap>[125X [27XSizeScreen( [ 72 ] );;[127X[104X
  [4X[32X[104X
  
  [33X[0;0YAgain  we  see  that  it  is  enough  to  consider  semidirect  products [22XG =
  2^10:M_22[122X, but this time for the two nonisomorphic modules.[133X
  
  [33X[0;0YWe  could  use the same method as in the first case for constructing the two
  groups.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xgens_1:= gens[1];;[127X[104X
    [4X[25Xgap>[125X [27Xmats:= List( [1 .. 3 ], x -> IdentityMat( d+1, GF(p) ) );;[127X[104X
    [4X[25Xgap>[125X [27Xv:= mats[1][ d+1 ];;[127X[104X
    [4X[25Xgap>[125X [27Xmats[1]{ [ 1 .. d ] }{ [ 1 .. d ] }:= gens_1[1];;[127X[104X
    [4X[25Xgap>[125X [27Xmats[2]{ [ 1 .. d ] }{ [ 1 .. d ] }:= gens_1[2];;[127X[104X
    [4X[25Xgap>[125X [27Xmats[3][ d+1 ][1]:= Z(p)^0;;[127X[104X
    [4X[25Xgap>[125X [27Xgrp_1:= Group( mats );;[127X[104X
    [4X[25Xgap>[125X [27XSize( grp_1 );[127X[104X
    [4X[28X454164480[128X[104X
    [4X[25Xgap>[125X [27Xgens_2:= gens[1];;[127X[104X
    [4X[25Xgap>[125X [27Xmats:= List( [1 .. 3 ], x -> IdentityMat( d+1, GF(p) ) );;[127X[104X
    [4X[25Xgap>[125X [27Xv:= mats[1][ d+1 ];;[127X[104X
    [4X[25Xgap>[125X [27Xmats[1]{ [ 1 .. d ] }{ [ 1 .. d ] }:= gens_2[1];;[127X[104X
    [4X[25Xgap>[125X [27Xmats[2]{ [ 1 .. d ] }{ [ 1 .. d ] }:= gens_2[2];;[127X[104X
    [4X[25Xgap>[125X [27Xmats[3][ d+1 ][1]:= Z(p)^0;;[127X[104X
    [4X[25Xgap>[125X [27Xgrp_2:= Group( mats );;[127X[104X
    [4X[25Xgap>[125X [27XSize( grp_2 );[127X[104X
    [4X[28X454164480[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  [5XGAP[105X  Character  Table Library contains the ordinary character tables of
  the  two  groups in question. We check this with the same approach as in the
  previous examples.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xiso:= IsomorphismTypeInfoFiniteSimpleGroup( s );[127X[104X
    [4X[28Xrec( name := "M(22)", series := "Spor", shortname := "M22" )[128X[104X
    [4X[25Xgap>[125X [27Xnames:= AllCharacterTableNames( Size, 2^10 * Size( s ) );;[127X[104X
    [4X[25Xgap>[125X [27Xcand:= List( names, CharacterTable );;[127X[104X
    [4X[25Xgap>[125X [27Xcand:= Filtered( cand,[127X[104X
    [4X[25X>[125X [27X     t -> ForAny( ClassPositionsOfMinimalNormalSubgroups( t ),[127X[104X
    [4X[25X>[125X [27X            n -> IsomorphismTypeInfoFiniteSimpleGroup( t / n ) = iso ) );[127X[104X
    [4X[28X[ CharacterTable( "2^10:M22'" ), CharacterTable( "2^10:m22" ) ][128X[104X
    [4X[25Xgap>[125X [27XList( cand, NrConjugacyClasses );[127X[104X
    [4X[28X[ 47, 43 ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YSo  we  can  easily  check  that  in  both  cases, [22XG[122X has two rational valued
  irreducibles of the degree [22X1155[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xt:= cand[1];;[127X[104X
    [4X[25Xgap>[125X [27Xrationals:= Filtered( Irr( t ), x -> IsSubset( Integers, x ) );;[127X[104X
    [4X[25Xgap>[125X [27XCollected( List( rationals, x -> x[1] ) );[127X[104X
    [4X[28X[ [ 1, 1 ], [ 21, 1 ], [ 22, 1 ], [ 55, 1 ], [ 99, 1 ], [ 154, 1 ], [128X[104X
    [4X[28X  [ 210, 1 ], [ 231, 3 ], [ 385, 1 ], [ 440, 1 ], [ 770, 5 ], [128X[104X
    [4X[28X  [ 924, 2 ], [ 1155, 2 ], [ 1386, 1 ], [ 1408, 1 ], [ 3080, 2 ], [128X[104X
    [4X[28X  [ 3465, 4 ], [ 4620, 2 ], [ 6930, 3 ], [ 9240, 1 ] ][128X[104X
    [4X[25Xgap>[125X [27Xt:= cand[2];;[127X[104X
    [4X[25Xgap>[125X [27Xrationals:= Filtered( Irr( t ), x -> IsSubset( Integers, x ) );;[127X[104X
    [4X[25Xgap>[125X [27XCollected( List( rationals, x -> x[1] ) );[127X[104X
    [4X[28X[ [ 1, 1 ], [ 21, 1 ], [ 55, 1 ], [ 77, 1 ], [ 99, 1 ], [ 154, 1 ], [128X[104X
    [4X[28X  [ 210, 1 ], [ 231, 1 ], [ 330, 1 ], [ 385, 3 ], [ 616, 2 ], [128X[104X
    [4X[28X  [ 693, 1 ], [ 770, 1 ], [ 1155, 2 ], [ 1980, 1 ], [ 2310, 4 ], [128X[104X
    [4X[28X  [ 2640, 1 ], [ 3465, 2 ], [ 4620, 1 ], [ 5544, 2 ], [ 6160, 1 ], [128X[104X
    [4X[28X  [ 6930, 2 ], [ 9856, 1 ] ][128X[104X
  [4X[32X[104X
  
  
  [1X10.3 [33X[0;0Y[22XG/N ≅ J_2[122X[101X[1X and [22X|N| = 2^12[122X[101X[1X[133X[101X
  
  [33X[0;0YThe group [22XS = J_2[122X has exactly one irreducible [22X12[122X-dimensional module over the
  field with two elements, up to isomorphism. This module can be obtained from
  any   of   the   two   absolutely  irreducible  [22X6[122X-dimensional  [22XS[122X-modules  in
  characteristic two, by regarding it as a module over the prime field [22XGF(2)[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xp:= 2;;  d:= 12;;[127X[104X
    [4X[25Xgap>[125X [27Xt:= CharacterTable( "J2" ) mod p;[127X[104X
    [4X[28XBrauerTable( "J2", 2 )[128X[104X
    [4X[25Xgap>[125X [27Xirr:= Filtered( Irr( t ), x -> x[1] <= d );;[127X[104X
    [4X[25Xgap>[125X [27XDisplay( t, rec( chars:= irr, powermap:= false,[127X[104X
    [4X[25X>[125X [27X                    centralizers:= false ) );[127X[104X
    [4X[28XJ2mod2[128X[104X
    [4X[28X[128X[104X
    [4X[28X       1a 3a 3b 5a 5b 5c 5d 7a 15a 15b[128X[104X
    [4X[28X[128X[104X
    [4X[28XY.1     1  1  1  1  1  1  1  1   1   1[128X[104X
    [4X[28XY.2     6 -3  .  A *A  B *B -1   C  *C[128X[104X
    [4X[28XY.3     6 -3  . *A  A *B  B -1  *C   C[128X[104X
    [4X[28X[128X[104X
    [4X[28XA = -2*E(5)-2*E(5)^4[128X[104X
    [4X[28X  = 1-Sqrt(5) = 1-r5[128X[104X
    [4X[28XB = E(5)+2*E(5)^2+2*E(5)^3+E(5)^4[128X[104X
    [4X[28X  = (-3-Sqrt(5))/2 = -2-b5[128X[104X
    [4X[28XC = E(5)+E(5)^4[128X[104X
    [4X[28X  = (-1+Sqrt(5))/2 = b5[128X[104X
    [4X[25Xgap>[125X [27XList( irr, x -> SizeOfFieldOfDefinition( x, p ) );[127X[104X
    [4X[28X[ 2, 4, 4 ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YFirst  we  construct the irreducible [22X12[122X-dimensional representation of [22XS[122X over
  [22XGF(2)[122X,  using  that  the  [5XAtlas[105X  of  Group  Representations  provides matrix
  generators for [22XS[122X in the [22X6[122X-dimensional representation over [22XGF(4)[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xinfo:= OneAtlasGeneratingSetInfo( "J2", Dimension, 6,[127X[104X
    [4X[25X>[125X [27X              Characteristic, p );[127X[104X
    [4X[28Xrec( charactername := "6a", constituents := [ 2 ], contents := "core",[128X[104X
    [4X[28X  dim := 6, groupname := "J2", id := "a", [128X[104X
    [4X[28X  identifier := [ "J2", [ "J2G1-f4r6aB0.m1", "J2G1-f4r6aB0.m2" ], 1, [128X[104X
    [4X[28X      4 ], repname := "J2G1-f4r6aB0", repnr := 16, ring := GF(2^2), [128X[104X
    [4X[28X  size := 604800, standardization := 1, type := "matff" )[128X[104X
    [4X[25Xgap>[125X [27Xgens_dim6:= AtlasGenerators( info ).generators;;[127X[104X
    [4X[25Xgap>[125X [27Xb:= Basis( GF(4) );;[127X[104X
    [4X[25Xgap>[125X [27Xgens_dim12:= List( gens_dim6, x -> BlownUpMatrix( b, x ) );;[127X[104X
  [4X[32X[104X
  
  [33X[0;0YWe claim that any extension of [22XS[122X with the given module splits.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xs:= AtlasGroup( "J2", IsPermGroup, true );;[127X[104X
    [4X[25Xgap>[125X [27Xchr:= CHR( s, p, 0, gens_dim12 );;[127X[104X
    [4X[25Xgap>[125X [27XSizeScreen( [ 100 ] );;[127X[104X
    [4X[25Xgap>[125X [27XSecondCohomologyDimension( chr );[127X[104X
    [4X[28X0[128X[104X
    [4X[25Xgap>[125X [27XSizeScreen( [ 72 ] );;[127X[104X
  [4X[32X[104X
  
  [33X[0;0YAgain  we  see  that  it  is  enough  to  consider  a semidirect product [22XG =
  2^12:J_2[122X.[133X
  
  [33X[0;0YHere is a description how we could construct the group.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xmats:= List( [ 1 .. 3 ], x -> IdentityMat( d+1, GF(p) ) );;[127X[104X
    [4X[25Xgap>[125X [27Xv:= mats[1][ d+1 ];;[127X[104X
    [4X[25Xgap>[125X [27Xmats[1]{ [ 1 .. d ] }{ [ 1 .. d ] }:= gens_dim12[1];;[127X[104X
    [4X[25Xgap>[125X [27Xmats[2]{ [ 1 .. d ] }{ [ 1 .. d ] }:= gens_dim12[2];;[127X[104X
    [4X[25Xgap>[125X [27Xmats[3][ d+1 ][1]:= Z(p)^0;;[127X[104X
    [4X[25Xgap>[125X [27Xgrp:= Group( mats );;[127X[104X
    [4X[25Xgap>[125X [27Xg:= Image( IsomorphismPermGroup( grp ) );;[127X[104X
    [4X[25Xgap>[125X [27XSize( g );[127X[104X
    [4X[28X2477260800[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  [5XGAP[105X Character Table Library contains the ordinary character table of [22XG[122X.
  We check this with the same approach as in the previous examples.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xiso:= IsomorphismTypeInfoFiniteSimpleGroup( s );[127X[104X
    [4X[28Xrec( name := "HJ = J(2) = F(5-)", series := "Spor", shortname := "J2" [128X[104X
    [4X[28X )[128X[104X
    [4X[25Xgap>[125X [27Xnames:= AllCharacterTableNames( Size, 2^12 * Size( s ) );;[127X[104X
    [4X[25Xgap>[125X [27Xcand:= List( names, CharacterTable );;[127X[104X
    [4X[25Xgap>[125X [27Xcand:= Filtered( cand,[127X[104X
    [4X[25X>[125X [27X     t -> ForAny( ClassPositionsOfMinimalNormalSubgroups( t ),[127X[104X
    [4X[25X>[125X [27X            n -> IsomorphismTypeInfoFiniteSimpleGroup( t / n ) = iso ) );[127X[104X
    [4X[28X[ CharacterTable( "2^12:J2" ) ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YSo  we  can  easily check that [22XG[122X has two rational valued irreducibles of the
  degree [22X1575[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xt:= cand[1];;[127X[104X
    [4X[25Xgap>[125X [27Xrationals:= Filtered( Irr( t ), x -> IsSubset( Integers, x ) );;[127X[104X
    [4X[25Xgap>[125X [27XCollected( List( rationals, x -> x[1] ) );[127X[104X
    [4X[28X[ [ 1, 1 ], [ 36, 1 ], [ 63, 1 ], [ 90, 1 ], [ 126, 1 ], [ 160, 1 ], [128X[104X
    [4X[28X  [ 175, 1 ], [ 225, 1 ], [ 288, 1 ], [ 300, 1 ], [ 336, 1 ], [128X[104X
    [4X[28X  [ 1575, 2 ], [ 2520, 4 ], [ 3150, 1 ], [ 4725, 6 ], [ 9450, 1 ], [128X[104X
    [4X[28X  [ 10080, 4 ], [ 12600, 4 ], [ 18900, 2 ] ][128X[104X
  [4X[32X[104X
  
  
  [1X10.4 [33X[0;0Y[22XG/N ≅ J_2[122X[101X[1X and [22X|N| = 5^14[122X[101X[1X[133X[101X
  
  [33X[0;0YThe group [22XS = J_2[122X has exactly one irreducible [22X14[122X-dimensional module over the
  field  with [22X5[122X elements, up to isomorphism. This module is in fact absolutely
  irreducible.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xp:= 5;;  d:= 14;;[127X[104X
    [4X[25Xgap>[125X [27Xt:= CharacterTable( "J2" ) mod p;[127X[104X
    [4X[28XBrauerTable( "J2", 5 )[128X[104X
    [4X[25Xgap>[125X [27Xirr:= Filtered( Irr( t ), x -> x[1] <= d );;[127X[104X
    [4X[25Xgap>[125X [27XDisplay( t, rec( chars:= irr, powermap:= false,[127X[104X
    [4X[25X>[125X [27X                    centralizers:= false ) );[127X[104X
    [4X[28XJ2mod5[128X[104X
    [4X[28X[128X[104X
    [4X[28X       1a 2a 2b 3a 3b 4a 6a 6b 7a 8a 12a[128X[104X
    [4X[28X[128X[104X
    [4X[28XY.1     1  1  1  1  1  1  1  1  1  1   1[128X[104X
    [4X[28XY.2    14 -2  2  5 -1  2  1 -1  .  .  -1[128X[104X
  [4X[32X[104X
  
  [33X[0;0YIn  this  case, we do not attempt to compute the complete character table of
  [22XG[122X.  Instead,  we  show that [22XG/N[122X has at least five regular orbits on the dual
  space  of  [22XN[122X,  and  apply \cite[Lemma 5.1 (i)]{DNT}. (Note that [22XN[122X is in fact
  self-dual.)[133X
  
  [33X[0;0YFor  that, we use [5XGAP[105X's table of marks of [22XS[122X. The information stored for this
  table of marks allows us to compute, for each class of subgroups [22XU[122X of [22XS[122X, the
  numbers  of  orbits  in  the  dual  space  of  [22XN[122X for which contain the point
  stabilizers in [22XS[122X are exactly the conjugates of [22XU[122X. The following [5XGAP[105X function
  takes  the table of marks [10Xtom[110X of [22XS[122X, a list [10Xmatgens[110X of matrices that describe
  the  action  of the generators of [22XS[122X on the vector space in question, and the
  size  [10Xq[110X  of  its  field  of  scalars.  The return value is a record with the
  components  [10Xfixed[110X (the vector of numbers of fixed points of the subgroups of
  [22XS[122X  on  the  dual of [22XN[122X), [10Xdecomp[110X (the numbers of orbits with the corresponding
  point  stabilizers),  [10Xnonzeropos[110X  (the  positions of subgroups that occur as
  point stabilizers), and [10Xstaborders[110X (the list of orders of the subgroups that
  occur as point stabilizers).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xorbits_from_tom:= function( tom, matgens, q )[127X[104X
    [4X[25X>[125X [27X    local slp, fixed, idmat, i, rest, decomp, nonzeropos;[127X[104X
    [4X[25X>[125X [27X[127X[104X
    [4X[25X>[125X [27X    slp:= StraightLineProgramsTom( tom );[127X[104X
    [4X[25X>[125X [27X    fixed:= [];[127X[104X
    [4X[25X>[125X [27X    idmat:= matgens[1]^0;[127X[104X
    [4X[25X>[125X [27X    for i in [ 1 .. Length( slp ) ] do[127X[104X
    [4X[25X>[125X [27X      if IsList( slp[i] ) then[127X[104X
    [4X[25X>[125X [27X        # Each subgroup generator has a program of its own.[127X[104X
    [4X[25X>[125X [27X        rest:= List( slp[i],[127X[104X
    [4X[25X>[125X [27X                     prg -> ResultOfStraightLineProgram( prg, gens ) );[127X[104X
    [4X[25X>[125X [27X      else[127X[104X
    [4X[25X>[125X [27X        # The subgroup generators are computed with one common program.[127X[104X
    [4X[25X>[125X [27X        rest:= ResultOfStraightLineProgram( slp[i], gens );[127X[104X
    [4X[25X>[125X [27X      fi;[127X[104X
    [4X[25X>[125X [27X      if IsEmpty( rest ) then[127X[104X
    [4X[25X>[125X [27X        # The subgroup is trivial.[127X[104X
    [4X[25X>[125X [27X        fixed[i]:= q^Length( idmat );[127X[104X
    [4X[25X>[125X [27X      else[127X[104X
    [4X[25X>[125X [27X        # Compute the intersection of fixed spaces of the transposed[127X[104X
    [4X[25X>[125X [27X        # matrices, since we act on Irr(N) not on N.[127X[104X
    [4X[25X>[125X [27X        fixed[i]:= q^Length( NullspaceMat( TransposedMat( Concatenation([127X[104X
    [4X[25X>[125X [27X                       List( rest, x -> x - idmat ) ) ) ) );[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    decomp:= DecomposedFixedPointVector( tom, fixed );[127X[104X
    [4X[25X>[125X [27X    nonzeropos:= Filtered( [ 1 .. Length( decomp ) ],[127X[104X
    [4X[25X>[125X [27X                           i -> decomp[i] <> 0 );[127X[104X
    [4X[25X>[125X [27X[127X[104X
    [4X[25X>[125X [27X    return rec( fixed:= fixed,[127X[104X
    [4X[25X>[125X [27X                decomp:= decomp,[127X[104X
    [4X[25X>[125X [27X                nonzeropos:= nonzeropos,[127X[104X
    [4X[25X>[125X [27X                staborders:= OrdersTom( tom ){ nonzeropos },[127X[104X
    [4X[25X>[125X [27X              );[127X[104X
    [4X[25X>[125X [27Xend;;[127X[104X
  [4X[32X[104X
  
  [33X[0;0YNote  that  this function assumes that the generators of [22XS[122X obtained from the
  [5XAtlas[105X of Group Representations are compatible with the generators from [5XGAP[105X's
  table  of  marks  of [22XS[122X. This fact can be read off from the [9Xtrue[109X value of the
  [10XATLAS[110X      component      in     the     [2XStandardGeneratorsInfo[102X     ([14XTomLib:
  StandardGeneratorsInfo for groups[114X) value of the table of marks.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xtom:= TableOfMarks( "J2" );[127X[104X
    [4X[28XTableOfMarks( "J2" )[128X[104X
    [4X[25Xgap>[125X [27XStandardGeneratorsInfo( tom );[127X[104X
    [4X[28X[ rec( ATLAS := true, [128X[104X
    [4X[28X      description := "|z|=10, z^5=a, |b|=3, |C(b)|=36, |ab|=7", [128X[104X
    [4X[28X      generators := "a, b", [128X[104X
    [4X[28X      script := [128X[104X
    [4X[28X        [ [ 1, 10, 5 ], [ 2, 3 ], [ [ 2, 1 ], [ "|C(",, ")|" ], 36 ], [128X[104X
    [4X[28X          [ 1, 1, 2, 1, 7 ] ], standardization := 1 ) ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YAlternatively,  we  can  compute  whether  the generators are compatible, as
  follows.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xinfo:= OneAtlasGeneratingSetInfo( "J2", Dimension, d, Ring, GF(p) );[127X[104X
    [4X[28Xrec( charactername := "14a", constituents := [ 2 ], [128X[104X
    [4X[28X  contents := "core", dim := 14, groupname := "J2", id := "", [128X[104X
    [4X[28X  identifier := [ "J2", [ "J2G1-f5r14B0.m1", "J2G1-f5r14B0.m2" ], 1, [128X[104X
    [4X[28X      5 ], repname := "J2G1-f5r14B0", repnr := 19, ring := GF(5), [128X[104X
    [4X[28X  size := 604800, standardization := 1, type := "matff" )[128X[104X
    [4X[25Xgap>[125X [27Xgens:= AtlasGenerators( info ).generators;;[127X[104X
    [4X[25Xgap>[125X [27Xmap:= GroupGeneralMappingByImages( UnderlyingGroup( tom ),[127X[104X
    [4X[25X>[125X [27X     Group( gens ), GeneratorsOfGroup( UnderlyingGroup( tom ) ), gens );;[127X[104X
    [4X[25Xgap>[125X [27XIsGroupHomomorphism( map );[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [33X[0;0YNow we are sure that we may apply the function [10Xorbits_from_tom[110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xorbits_from_tom( tom, gens, p );[127X[104X
    [4X[28Xrec( [128X[104X
    [4X[28X  decomp := [ 8600, 0, 2512, 359, 10, 0, 0, 212, 5, 0, 0, 4, 0, 240, [128X[104X
    [4X[28X      16, 10, 0, 0, 0, 0, 10, 0, 0, 0, 0, 2, 0, 0, 36, 0, 0, 0, 26, [128X[104X
    [4X[28X      0, 0, 0, 0, 0, 0, 0, 20, 0, 10, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0, [128X[104X
    [4X[28X      0, 0, 0, 0, 10, 0, 0, 5, 0, 0, 0, 26, 0, 10, 0, 0, 0, 0, 10, 0, [128X[104X
    [4X[28X      0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10, 0, 0, 0, 2, 0, [128X[104X
    [4X[28X      0, 0, 0, 2, 4, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, [128X[104X
    [4X[28X      16, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, [128X[104X
    [4X[28X      0, 0, 4, 0, 0, 0, 4, 0, 0, 1 ], [128X[104X
    [4X[28X  fixed := [ 6103515625, 15625, 390625, 390625, 625, 25, 3125, 3125, [128X[104X
    [4X[28X      625, 625, 625, 625, 5, 3125, 125, 625, 25, 25, 125, 5, 125, 25, [128X[104X
    [4X[28X      125, 25, 25, 25, 5, 125, 125, 125, 25, 25, 3125, 1, 1, 5, 5, [128X[104X
    [4X[28X      25, 5, 25, 125, 5, 25, 25, 25, 25, 25, 25, 5, 25, 25, 5, 25, 5, [128X[104X
    [4X[28X      5, 5, 5, 25, 25, 1, 125, 1, 5, 5, 125, 1, 25, 5, 25, 1, 5, 25, [128X[104X
    [4X[28X      5, 5, 25, 25, 5, 5, 5, 1, 5, 5, 1, 1, 1, 5, 1, 25, 25, 25, 1, [128X[104X
    [4X[28X      5, 25, 5, 5, 1, 1, 125, 5, 5, 5, 25, 5, 5, 5, 1, 1, 5, 5, 1, 5, [128X[104X
    [4X[28X      1, 5, 1, 1, 25, 5, 5, 1, 1, 1, 1, 5, 1, 1, 25, 1, 1, 5, 1, 1, [128X[104X
    [4X[28X      5, 1, 5, 1, 1, 5, 1, 5, 1, 1, 1, 5, 1, 1, 1 ], [128X[104X
    [4X[28X  nonzeropos := [ 1, 3, 4, 5, 8, 9, 12, 14, 15, 16, 21, 26, 29, 33, [128X[104X
    [4X[28X      41, 43, 44, 58, 61, 65, 67, 72, 89, 93, 98, 99, 105, 116, 126, [128X[104X
    [4X[28X      139, 143, 146 ], [128X[104X
    [4X[28X  staborders := [ 1, 2, 3, 3, 4, 4, 5, 6, 6, 6, 8, 9, 10, 12, 12, 12, [128X[104X
    [4X[28X      14, 20, 24, 24, 24, 30, 48, 50, 60, 60, 72, 120, 192, 600, [128X[104X
    [4X[28X      1920, 604800 ] )[128X[104X
  [4X[32X[104X
  
  [33X[0;0YWe see that [22XS[122X has [22X8600[122X regular orbits on (the dual space of) [22XN[122X.[133X
  
  
  [1X10.5 [33X[0;0Y[22XG/N ≅ J_2[122X[101X[1X and [22X|N| = 2^28[122X[101X[1X[133X[101X
  
  [33X[0;0YThe group [22XS = J_2[122X has exactly one irreducible [22X28[122X-dimensional module over the
  field with two elements, up to isomorphism. This module can be obtained from
  any   of   the   two  absolutely  irreducible  [22X14[122X-dimensional  [22XS[122X-modules  in
  characteristic two, by regarding it as a module over the prime field [22XGF(2)[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xp:= 2;;  d:= 28;;[127X[104X
    [4X[25Xgap>[125X [27Xt:= CharacterTable( "J2" ) mod p;[127X[104X
    [4X[28XBrauerTable( "J2", 2 )[128X[104X
    [4X[25Xgap>[125X [27Xirr:= Filtered( Irr( t ), x -> x[1] <= d );;[127X[104X
    [4X[25Xgap>[125X [27XDisplay( t, rec( chars:= irr, powermap:= false,[127X[104X
    [4X[25X>[125X [27X                    centralizers:= false ) );[127X[104X
    [4X[28XJ2mod2[128X[104X
    [4X[28X[128X[104X
    [4X[28X       1a 3a 3b 5a 5b  5c  5d 7a 15a 15b[128X[104X
    [4X[28X[128X[104X
    [4X[28XY.1     1  1  1  1  1   1   1  1   1   1[128X[104X
    [4X[28XY.2     6 -3  .  A *A   C  *C -1   D  *D[128X[104X
    [4X[28XY.3     6 -3  . *A  A  *C   C -1  *D   D[128X[104X
    [4X[28XY.4    14  5 -1  B *B  -C -*C  .   .   .[128X[104X
    [4X[28XY.5    14  5 -1 *B  B -*C  -C  .   .   .[128X[104X
    [4X[28X[128X[104X
    [4X[28XA = -2*E(5)-2*E(5)^4[128X[104X
    [4X[28X  = 1-Sqrt(5) = 1-r5[128X[104X
    [4X[28XB = -3*E(5)-3*E(5)^4[128X[104X
    [4X[28X  = (3-3*Sqrt(5))/2 = -3b5[128X[104X
    [4X[28XC = E(5)+2*E(5)^2+2*E(5)^3+E(5)^4[128X[104X
    [4X[28X  = (-3-Sqrt(5))/2 = -2-b5[128X[104X
    [4X[28XD = E(5)+E(5)^4[128X[104X
    [4X[28X  = (-1+Sqrt(5))/2 = b5[128X[104X
    [4X[25Xgap>[125X [27XList( irr, x -> SizeOfFieldOfDefinition( x, p ) );[127X[104X
    [4X[28X[ 2, 4, 4, 4, 4 ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YWe use the same approach as in the previous example.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xtom:= TableOfMarks( "J2" );;[127X[104X
    [4X[25Xgap>[125X [27Xinfo:= OneAtlasGeneratingSetInfo( "J2", Dimension, 14, Ring, GF(4) );;[127X[104X
    [4X[25Xgap>[125X [27Xgens:= List( AtlasGenerators( info ).generators,[127X[104X
    [4X[25X>[125X [27X                x -> BlownUpMat( Basis(GF(4)), x ) );;[127X[104X
    [4X[25Xgap>[125X [27Xorbits_from_tom( tom, gens, p );[127X[104X
    [4X[28Xrec( [128X[104X
    [4X[28X  decomp := [ 235, 33, 282, 38, 0, 0, 6, 31, 36, 0, 0, 0, 3, 66, 9, [128X[104X
    [4X[28X      0, 0, 0, 0, 0, 0, 2, 18, 0, 0, 1, 0, 0, 15, 0, 0, 0, 6, 0, 0, [128X[104X
    [4X[28X      0, 0, 0, 0, 0, 12, 0, 0, 5, 0, 1, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, [128X[104X
    [4X[28X      0, 0, 0, 0, 3, 1, 3, 0, 9, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, [128X[104X
    [4X[28X      0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, [128X[104X
    [4X[28X      3, 0, 0, 0, 0, 0, 3, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 9, 0, 0, 0, [128X[104X
    [4X[28X      0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 3, 0, [128X[104X
    [4X[28X      0, 0, 3, 0, 0, 1 ], [128X[104X
    [4X[28X  fixed := [ 268435456, 65536, 65536, 65536, 256, 1024, 4096, 1024, [128X[104X
    [4X[28X      1024, 256, 256, 256, 64, 1024, 64, 256, 16, 16, 64, 64, 64, [128X[104X
    [4X[28X      256, 256, 64, 16, 16, 64, 64, 64, 64, 16, 16, 1024, 4, 4, 4, 4, [128X[104X
    [4X[28X      16, 16, 16, 64, 16, 16, 16, 16, 64, 16, 16, 16, 64, 16, 16, 16, [128X[104X
    [4X[28X      16, 4, 16, 16, 16, 16, 1, 64, 4, 16, 4, 64, 4, 16, 4, 16, 1, 4, [128X[104X
    [4X[28X      16, 4, 4, 16, 16, 4, 4, 16, 1, 4, 16, 1, 1, 1, 16, 4, 16, 16, [128X[104X
    [4X[28X      16, 1, 4, 16, 4, 4, 1, 4, 64, 4, 4, 4, 16, 4, 4, 4, 1, 1, 4, [128X[104X
    [4X[28X      16, 1, 4, 1, 4, 1, 4, 16, 4, 4, 1, 1, 1, 1, 4, 1, 1, 16, 1, 1, [128X[104X
    [4X[28X      4, 1, 4, 4, 1, 4, 1, 1, 4, 1, 4, 1, 1, 1, 4, 1, 1, 1 ], [128X[104X
    [4X[28X  nonzeropos := [ 1, 2, 3, 4, 7, 8, 9, 13, 14, 15, 22, 23, 26, 29, [128X[104X
    [4X[28X      33, 41, 44, 46, 50, 61, 62, 63, 65, 72, 82, 93, 99, 105, 109, [128X[104X
    [4X[28X      116, 126, 131, 139, 143, 146 ], [128X[104X
    [4X[28X  staborders := [ 1, 2, 2, 3, 4, 4, 4, 6, 6, 6, 8, 8, 9, 10, 12, 12, [128X[104X
    [4X[28X      14, 16, 16, 24, 24, 24, 24, 30, 40, 50, 60, 72, 96, 120, 192, [128X[104X
    [4X[28X      240, 600, 1920, 604800 ] )[128X[104X
  [4X[32X[104X
  
  [33X[0;0YWe see that [22XS[122X has [22X235[122X regular orbits on (the dual space of) [22XN[122X.[133X
  
  
  [1X10.6 [33X[0;0Y[22XG/N ≅ ^3D_4(2)[122X[101X[1X and [22X|N| = 2^26[122X[101X[1X[133X[101X
  
  [33X[0;0YThe  group  [22XS  =  ^3D_4(2)[122X has exactly one irreducible [22X26[122X-dimensional module
  over  the field with two elements, up to isomorphism. This module is in fact
  absolutely irreducible.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xp:= 2;;  d:= 26;;[127X[104X
    [4X[25Xgap>[125X [27Xt:= CharacterTable( "3D4(2)" ) mod p;[127X[104X
    [4X[28XBrauerTable( "3D4(2)", 2 )[128X[104X
    [4X[25Xgap>[125X [27Xirr:= Filtered( Irr( t ), x -> x[1] <= d );;[127X[104X
    [4X[25Xgap>[125X [27XDisplay( t, rec( chars:= irr, powermap:= false,[127X[104X
    [4X[25X>[125X [27X                    centralizers:= false ) );[127X[104X
    [4X[28X3D4(2)mod2[128X[104X
    [4X[28X[128X[104X
    [4X[28X       1a 3a 3b 7a 7b 7c 7d 9a 9b 9c 13a 13b 13c 21a 21b 21c[128X[104X
    [4X[28X[128X[104X
    [4X[28XY.1     1  1  1  1  1  1  1  1  1  1   1   1   1   1   1   1[128X[104X
    [4X[28XY.2     8  2 -1  A  C  B  1  D  F  E   G   I   H   J   L   K[128X[104X
    [4X[28XY.3     8  2 -1  B  A  C  1  E  D  F   H   G   I   K   J   L[128X[104X
    [4X[28XY.4     8  2 -1  C  B  A  1  F  E  D   I   H   G   L   K   J[128X[104X
    [4X[28XY.5    26 -1 -1  5  5  5 -2  2  2  2   .   .   .  -1  -1  -1[128X[104X
    [4X[28X[128X[104X
    [4X[28XA = 3*E(7)^2+E(7)^3+E(7)^4+3*E(7)^5[128X[104X
    [4X[28XB = 3*E(7)+E(7)^2+E(7)^5+3*E(7)^6[128X[104X
    [4X[28XC = E(7)+3*E(7)^3+3*E(7)^4+E(7)^6[128X[104X
    [4X[28XD = -E(9)^2+E(9)^3-2*E(9)^4-2*E(9)^5+E(9)^6-E(9)^7[128X[104X
    [4X[28XE = -E(9)^2+E(9)^3+E(9)^4+E(9)^5+E(9)^6-E(9)^7[128X[104X
    [4X[28XF = 2*E(9)^2+E(9)^3+E(9)^4+E(9)^5+E(9)^6+2*E(9)^7[128X[104X
    [4X[28XG = E(13)+E(13)^2+E(13)^3+E(13)^5+E(13)^8+E(13)^10+E(13)^11+E(13)^12[128X[104X
    [4X[28XH = E(13)+E(13)^4+E(13)^5+E(13)^6+E(13)^7+E(13)^8+E(13)^9+E(13)^12[128X[104X
    [4X[28XI = E(13)^2+E(13)^3+E(13)^4+E(13)^6+E(13)^7+E(13)^9+E(13)^10+E(13)^11[128X[104X
    [4X[28XJ = E(7)^3+E(7)^4[128X[104X
    [4X[28XK = E(7)^2+E(7)^5[128X[104X
    [4X[28XL = E(7)+E(7)^6[128X[104X
  [4X[32X[104X
  
  [33X[0;0YWe try the same approach as in the examples about the group [22XJ_2[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xtom:= TableOfMarks( "3D4(2)" );[127X[104X
    [4X[28XTableOfMarks( "3D4(2)" )[128X[104X
    [4X[25Xgap>[125X [27XStandardGeneratorsInfo( tom );[127X[104X
    [4X[28X[ rec( ATLAS := true, [128X[104X
    [4X[28X      description := "|z|=8, z^4=a, |b|=9, |ab|=13, |abb|=8", [128X[104X
    [4X[28X      generators := "a, b", [128X[104X
    [4X[28X      script := [ [ 1, 8, 4 ], [ 2, 9 ], [ 1, 1, 2, 1, 13 ], [128X[104X
    [4X[28X          [ 1, 1, 2, 1, 2, 1, 8 ] ], standardization := 1 ) ][128X[104X
    [4X[25Xgap>[125X [27Xinfo:= OneAtlasGeneratingSetInfo( "3D4(2)", Dimension, 26, Ring, GF(2) );;[127X[104X
    [4X[25Xgap>[125X [27Xgens:= AtlasGenerators( info ).generators;;[127X[104X
    [4X[25Xgap>[125X [27Xmap:= GroupGeneralMappingByImages( UnderlyingGroup( tom ),[127X[104X
    [4X[25X>[125X [27X     Group( gens ), GeneratorsOfGroup( UnderlyingGroup( tom ) ), gens );;[127X[104X
    [4X[25Xgap>[125X [27XIsGroupHomomorphism( map );[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [33X[0;0YNow we apply the function [10Xorbits_from_tom[110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xorbsinfo:= orbits_from_tom( tom, gens, p );;[127X[104X
    [4X[25Xgap>[125X [27Xorbsinfo.fixed[1];[127X[104X
    [4X[28X67108864[128X[104X
    [4X[25Xgap>[125X [27Xorbsinfo.decomp[1];[127X[104X
    [4X[28X0[128X[104X
  [4X[32X[104X
  
  [33X[0;0YUnfortunately,  [22XS[122X has no regular orbit on (the dual of) [22XN[122X. However, there is
  one orbit whose point stabilizer in [22XS[122X is a dihedral group [22XD_18[122X of order [22X18[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xorbsinfo.staborders;[127X[104X
    [4X[28X[ 16, 16, 18, 42, 48, 52, 64, 72, 392, 1008, 1536, 3024, 3072, 3584, [128X[104X
    [4X[28X  258048, 211341312 ][128X[104X
    [4X[25Xgap>[125X [27Xorbsinfo.nonzeropos[3];[127X[104X
    [4X[28X446[128X[104X
    [4X[25Xgap>[125X [27Xorbsinfo.decomp[446];[127X[104X
    [4X[28X1[128X[104X
    [4X[25Xgap>[125X [27Xu:= RepresentativeTom( tom, 446 );[127X[104X
    [4X[28X<permutation group of size 18 with 2 generators>[128X[104X
    [4X[25Xgap>[125X [27XIsDihedralGroup( u );[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThus  there  ia  a linear character [22Xλ[122X of [22XN[122X whose inertia subgroup [22XT = I_G(λ)[122X
  has  the  structure  [22XN.D_18[122X.  Now  [22XIrr( T | λ )[122X can be identified with those
  irreducibles  of  [22XT/ker(λ)[122X that restrict nontrivially to [22XN/ker(λ)[122X, and there
  are only two groups, up to isomorphism, that can occur as [22XT/ker(λ)[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xcand:= Filtered( AllSmallGroups( 36 ),[127X[104X
    [4X[25X>[125X [27X            x -> Size( Centre( x ) ) = 2 and[127X[104X
    [4X[25X>[125X [27X                 IsDihedralGroup( x / Centre( x ) ) );[127X[104X
    [4X[28X[ <pc group of size 36 with 4 generators>, [128X[104X
    [4X[28X  <pc group of size 36 with 4 generators> ][128X[104X
    [4X[25Xgap>[125X [27XList( cand, StructureDescription );[127X[104X
    [4X[28X[ "C9 : C4", "D36" ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YThese two groups are a split and a nonsplit extension of the cyclic group of
  order  [22X18[122X  with a group of order two that acts by inverting. In other words,
  these two groups are the direct product of [22XD_18[122X with a cyclic group of order
  two and the subdirect product of [22XD_18[122X with a cyclic group of order four.[133X
  
  [33X[0;0YBoth  groups  possess  irreducible  characters  of  degree two, one rational
  valued and the other not, which restrict nontrivially to the centre.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XDisplay( CharacterTable( "Dihedral", 18 ) );[127X[104X
    [4X[28XDihedral(18)[128X[104X
    [4X[28X[128X[104X
    [4X[28X     2  1  .  .  .  .  1[128X[104X
    [4X[28X     3  2  2  2  2  2  .[128X[104X
    [4X[28X[128X[104X
    [4X[28X       1a 9a 9b 3a 9c 2a[128X[104X
    [4X[28X    2P 1a 9b 9c 3a 9a 1a[128X[104X
    [4X[28X    3P 1a 3a 3a 1a 3a 2a[128X[104X
    [4X[28X[128X[104X
    [4X[28XX.1     1  1  1  1  1  1[128X[104X
    [4X[28XX.2     1  1  1  1  1 -1[128X[104X
    [4X[28XX.3     2  A  B -1  C  .[128X[104X
    [4X[28XX.4     2  B  C -1  A  .[128X[104X
    [4X[28XX.5     2 -1 -1  2 -1  .[128X[104X
    [4X[28XX.6     2  C  A -1  B  .[128X[104X
    [4X[28X[128X[104X
    [4X[28XA = -E(9)^2-E(9)^4-E(9)^5-E(9)^7[128X[104X
    [4X[28XB = E(9)^2+E(9)^7[128X[104X
    [4X[28XC = E(9)^4+E(9)^5[128X[104X
  [4X[32X[104X
  
  [33X[0;0YBy \cite[Lemma 5.1 (ii)]{DNT}, we are done.[133X
  
  
  [1X10.7 [33X[0;0Y[22XG/N ≅ ^3D_4(2)[122X[101X[1X and [22X|N| = 3^25[122X[101X[1X[133X[101X
  
  [33X[0;0YThe  group  [22XS  =  ^3D_4(2)[122X has exactly one irreducible [22X25[122X-dimensional module
  over  the  field  with  three elements, up to isomorphism. This module is in
  fact absolutely irreducible.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xp:= 3;;  d:= 25;;[127X[104X
    [4X[25Xgap>[125X [27Xt:= CharacterTable( "3D4(2)" ) mod p;[127X[104X
    [4X[28XBrauerTable( "3D4(2)", 3 )[128X[104X
    [4X[25Xgap>[125X [27Xirr:= Filtered( Irr( t ), x -> x[1] <= d );;[127X[104X
    [4X[25Xgap>[125X [27XDisplay( t, rec( chars:= irr, powermap:= false,[127X[104X
    [4X[25X>[125X [27X                    centralizers:= false ) );[127X[104X
    [4X[28X3D4(2)mod3[128X[104X
    [4X[28X[128X[104X
    [4X[28X       1a 2a 2b 4a 4b 4c 7a 7b 7c 7d 8a 8b 13a 13b 13c 14a 14b 14c 28a[128X[104X
    [4X[28X[128X[104X
    [4X[28XY.1     1  1  1  1  1  1  1  1  1  1  1  1   1   1   1   1   1   1   1[128X[104X
    [4X[28XY.2    25 -7  1  5 -3  1  4  4  4 -3 -1 -1  -1  -1  -1   .   .   .  -2[128X[104X
    [4X[28X[128X[104X
    [4X[28X       28b 28c[128X[104X
    [4X[28X[128X[104X
    [4X[28XY.1      1   1[128X[104X
    [4X[28XY.2     -2  -2[128X[104X
  [4X[32X[104X
  
  [33X[0;0YWe use the same approach as in the examples about the group [22XJ_2[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xtom:= TableOfMarks( "3D4(2)" );;[127X[104X
    [4X[25Xgap>[125X [27Xinfo:= OneAtlasGeneratingSetInfo( "3D4(2)", Dimension, d, Ring, GF(p) );;[127X[104X
    [4X[25Xgap>[125X [27Xgens:= AtlasGenerators( info ).generators;;[127X[104X
    [4X[25Xgap>[125X [27Xorbsinfo:= orbits_from_tom( tom, gens, p );;[127X[104X
    [4X[25Xgap>[125X [27Xorbsinfo.fixed[1];[127X[104X
    [4X[28X847288609443[128X[104X
    [4X[25Xgap>[125X [27Xorbsinfo.decomp[1];[127X[104X
    [4X[28X3551[128X[104X
  [4X[32X[104X
  
  [33X[0;0YWe see that [22XS[122X has [22X3551[122X regular orbits on (the dual space of) [22XN[122X.[133X
  
