  
  [1X2 [33X[0;0YIrreducible Representations[133X[101X
  
  [33X[0;0YLet  [22XG[122X be a finite group and [22Xχ[122X be an ordinary irreducible character of [22XG[122X. In
  this   chapter   we   introduce   some  functions  to  construct  a  complex
  representation  [22XR[122X  of  [22XG[122X  affording  [22Xχ[122X. We proceed recursively, reducing the
  problem  to  smaller subgroups of [22XG[122X or characters of smaller degree until we
  obtain  a  problem  which  we  can deal with directly. Inputs of most of the
  functions are a given group [22XG[122X, and an irreducible character [22Xχ[122X. The output is
  a  mapping  (representation) which assigns to each generator [22Xx[122X of [22XG[122X a matrix
  [22XR(x)[122X.  We  can  use  these  functions  for  all  groups  and all irreducible
  characters  [22Xχ[122X of degree less than 100 although in principle the same methods
  can  be  extended  to characters of larger degree. The main methods in these
  functions  which  are used to construct representations of finite groups are
  Induction,  Extension,  Tensor  Product and Dixon's method (for constructing
  representations  of  simple  groups and their covers) [DA05], and Projective
  Representation method [DD10].[133X
  
  
  [1X2.1 [33X[0;0YConstructing Representations[133X[101X
  
  [33X[0;0YThis  section  introduces the main function to compute a representation of a
  finite group [22XG[122X affording an irreducible character [22Xχ[122X of [22XG[122X.[133X
  
  [1X2.1-1 IrreducibleAffordingRepresentation[101X
  
  [33X[1;0Y[29X[2XIrreducibleAffordingRepresentation[102X( [3Xchi[103X ) [32X function[133X
  
  [33X[0;0Ycalled with an irreducible character [3Xchi[103X of a group [22XG[122X, this function returns
  a  mapping  (representation) which maps each generator of [22XG[122X to a [22Xd*d[122X matrix,
  where  [22Xd[122X  is  the  degree of [3Xchi[103X. The group generated by these matrices (the
  image  of  the  map)  is  a matrix group which is isomorphic to [22XG[122X modulo the
  kernel  of the map. If [22XG[122X is a solvable group then there is no restriction on
  the  degree of [3Xchi[103X. In the case that [22XG[122X is not solvable and the character [3Xchi[103X
  has  degree  bigger  than  100 the output maybe is not correct. In this case
  sometimes  the output mapping does not afford the given character or it does
  not return any mapping.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xs := PerfectGroup( 129024, 2 );;[127X[104X
    [4X[25Xgap>[125X [27XG := Image(IsomorphismPermGroup( s ));;[127X[104X
    [4X[25Xgap>[125X [27Xchi := Irr( G )[36];; [127X[104X
    [4X[25Xgap>[125X [27Xchi[1];[127X[104X
    [4X[28X64[128X[104X
    [4X[25Xgap>[125X [27XIrreducibleAffordingRepresentation( chi );; [127X[104X
    [4X[28X#I  Warning: EpimorphismSchurCover via Holt's algorithm is under construction[128X[104X
    [4X[25Xgap>[125X [27Xtime; [127X[104X
    [4X[28X92657[128X[104X
    [4X[28X [128X[104X
  [4X[32X[104X
  
  [1X2.1-2 IsAffordingRepresentation[101X
  
  [33X[1;0Y[29X[2XIsAffordingRepresentation[102X( [3Xchi[103X, [3Xrep[103X ) [32X function[133X
  
  [33X[0;0YIf  [3Xchi[103X  and  [3Xrep[103X  are  a  character  and  a  representation  of  a group [22XG[122X,
  respectively,  then  [10XIsAffordingRepresentation[110X  returns [10Xtrue[110X if the trace of
  [3Xrep(x)[103X equals [3Xchi(x)[103X for all elements [22Xx[122X in [22XG[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XG := GL(2,7);;[127X[104X
    [4X[25Xgap>[125X [27Xchi := Irr(G)[ 29 ];;[127X[104X
    [4X[25Xgap>[125X [27Xrep := IrreducibleAffordingRepresentation( chi );[127X[104X
    [4X[28XCompositionMapping( [(8,15,22,29,36,43)(9,16,23,30,37,44)[128X[104X
    [4X[28X(10,17,24,31,38,45)(11,18,25,32,39,46)(12,19,26,33,40,47)[128X[104X
    [4X[28X(13,20,27,34,41,48)(14,21,28,35,42,49), (2,29,12)(3,36,20)[128X[104X
    [4X[28X(4,43,28)(5,8,30)(6,15,38)(7,22,46)(9,44,14)(10,16,17)[128X[104X
    [4X[28X(11,37,27)(13,23,39)(18,24,25)(19,45,35)(21,31,47)[128X[104X
    [4X[28X(26,32,33)(34,40,41)(42,48,49) ] ->[128X[104X
    [4X[28X[ [ [ 0, 0, 0, -1, 0, 0, 0 ],[128X[104X
    [4X[28X    [ 1, 0, -1, -1, 1, 0, -1 ] [128X[104X
    [4X[28X    [ 2, -1, -2, -2, 1, 2, -1 ],[128X[104X
    [4X[28X    [ 0, 0, -1, 0, 0, 0, 0 ],[128X[104X
    [4X[28X    [ 1, 0, -2, 0, 0, 1, -1 ],[128X[104X
    [4X[28X    [ 1, 0, -2, -1, 1, 1, -1 ],[128X[104X
    [4X[28X    [ -2, 1, 1, 1, -1, -1, 0 ] ],[128X[104X
    [4X[28X  [ [ 1, -1, -1, -1, 0, 2, -1 ],[128X[104X
    [4X[28X    [ 0, 0, 1, 0, 0, 0, 0 ],[128X[104X
    [4X[28X    [ 0, 0, 0, 0, 0, 1, 0 ],[128X[104X
    [4X[28X    [ 0, 1, -1, 0, 0, 0, -1 ],[128X[104X
    [4X[28X    [ 0, 1, 0, 1, 0, -1, 0 ],[128X[104X
    [4X[28X    [ 0, 1, 0, 0, 0, 0, 0 ],[128X[104X
    [4X[28X    [ 0, 0, 0, 0, -1, 0, 0 ] ] ], (action isomorphism) )[128X[104X
    [4X[25Xgap>[125X [27XIsAffordingRepresentation( chi, rep );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[28X [128X[104X
  [4X[32X[104X
  
  [33X[0;0YWe   can   obtain   the   size  of  the  image  of  this  representation  by
  [10XSize(Image(rep))[110X  and  compute  the value for an arbitrary element [22Xx[122X in [22XG[122X by
  [10Xx[110X^[10Xrep[110X.[133X
  
  
  [1X2.2 [33X[0;0YInduction[133X[101X
  
  [1X2.2-1 InducedSubgroupRepresentation[101X
  
  [33X[1;0Y[29X[2XInducedSubgroupRepresentation[102X( [3XG[103X, [3Xrep[103X ) [32X function[133X
  
  [33X[0;0Ycomputes  a  representation  of  [3XG[103X  induced from the representation [3Xrep[103X of a
  subgroup  [22XH[122X  of  [3XG[103X.  If  [3Xrep[103X  has  degree  [22Xd[122X  then  the degree of the output
  representation is [22Xd*|G:H|[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XG := SymmetricGroup( 6 );;[127X[104X
    [4X[25Xgap>[125X [27XH := AlternatingGroup( 6 );;[127X[104X
    [4X[25Xgap>[125X [27Xchi := Irr( H )[ 2 ];;[127X[104X
    [4X[25Xgap>[125X [27Xrep := IrreducibleAffordingRepresentation( chi );;[127X[104X
    [4X[25Xgap>[125X [27XInducedSubgroupRepresentation( G, rep ); [127X[104X
    [4X[28X[ (1,2,3,4,5,6), (1,2) ] ->[128X[104X
    [4X[28X[ [ [ 0, 0, 0, 0, 0, 1, 1, -1, -1, -1 ],[128X[104X
    [4X[28X    [ 0, 0, 0, 0, 0, 1, 0, -1, 0, -1 ],[128X[104X
    [4X[28X    [ 0, 0, 0, 0, 0, 1, 0, 0, -1, -1 ],[128X[104X
    [4X[28X    [ 0, 0, 0, 0, 0, 1, 0, 0, 0, 0 ],[128X[104X
    [4X[28X    [ 0, 0, 0, 0, 0, 0, 1, -1, 0, -1 ],[128X[104X
    [4X[28X    [ 1, 1, -1, -1, -1, 0, 0, 0, 0, 0 ],[128X[104X
    [4X[28X    [ 1, 0, 0, -1, -1, 0, 0, 0, 0, 0 ],[128X[104X
    [4X[28X    [ 1, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],[128X[104X
    [4X[28X    [ 1, 0, -1, 0, -1, 0, 0, 0, 0, 0 ],[128X[104X
    [4X[28X    [ 0, 1, 0, -1, -1, 0, 0, 0, 0, 0 ] ],[128X[104X
    [4X[28X  [ [ 0, 0, 0, 0, 0, 1, 0, 0, 0, 0 ],[128X[104X
    [4X[28X    [ 0, 0, 0, 0, 0, 0, 1, 0, 0, 0 ],[128X[104X
    [4X[28X    [ 0, 0, 0, 0, 0, 0, 0, 0, 1, 0 ],[128X[104X
    [4X[28X    [ 0, 0, 0, 0, 0, 0, 0, 1, 0, 0 ],[128X[104X
    [4X[28X    [ 0, 0, 0, 0, 0, 1, 1, -1, -1, -1 ],[128X[104X
    [4X[28X    [ 1, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],[128X[104X
    [4X[28X    [ 0, 1, 0, 0, 0, 0, 0, 0, 0, 0 ],[128X[104X
    [4X[28X    [ 0, 0, 0, 1, 0, 0, 0, 0, 0, 0 ],[128X[104X
    [4X[28X    [ 0, 0, 1, 0, 0, 0, 0, 0, 0, 0 ],[128X[104X
    [4X[28X    [ 1, 1, -1, -1, -1, 0, 0, 0, 0, 0 ] ] ][128X[104X
    [4X[28X	[128X[104X
  [4X[32X[104X
  
  
  [1X2.3 [33X[0;0YExtension[133X[101X
  
  [33X[0;0YIn  this  section we introduce some functions for extending a representation
  of a subgroup to the whole group.[133X
  
  [1X2.3-1 ExtendedRepresentation[101X
  
  [33X[1;0Y[29X[2XExtendedRepresentation[102X( [3Xchi[103X, [3Xrep[103X ) [32X function[133X
  
  [33X[0;0YSuppose  [22XH[122X is a subgroup of a group [22XG[122X and [3Xchi[103X is an irreducible character of
  [22XG[122X  such that the restriction of [3Xchi[103X to [22XH[122X, [22Xphi[122X say, is irreducible. If [3Xrep[103X is
  an irreducible representation of [22XH[122X affording [22Xphi[122X then [10XExtendedRepresentation[110X
  extends  the representation [3Xrep[103X of [22XH[122X to a representation of [22XG[122X affording [3Xchi[103X.
  This  function call can be quite expensive when the representation [3Xrep[103X has a
  large degree.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XG := AlternatingGroup( 6 );;[127X[104X
    [4X[25Xgap>[125X [27XH := Group([ (1,2,3,4,6), (1,4)(5,6) ]);;[127X[104X
    [4X[25Xgap>[125X [27Xchi := Irr( G )[ 2 ];;[127X[104X
    [4X[25Xgap>[125X [27Xphi := RestrictedClassFunction( chi, H );;[127X[104X
    [4X[25Xgap>[125X [27XIsIrreducibleCharacter( phi );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xrep := IrreducibleAffordingRepresentation( phi );;[127X[104X
    [4X[25Xgap>[125X [27Xext := ExtendedRepresentation( chi, rep );[127X[104X
    [4X[28X#I  Need to extend a representation of degree 5. This may take a while.[128X[104X
    [4X[28X[ (1,2,3,4,5), (4,5,6) ] -> [[128X[104X
    [4X[28X[ [ 0, 1, 0, -1, -1 ],[128X[104X
    [4X[28X  [ 0, 0, 0, 1, 0 ],[128X[104X
    [4X[28X  [ -1, -1, -1, 0, 0 ],[128X[104X
    [4X[28X  [ 0, 0, 0, 0, -1 ],[128X[104X
    [4X[28X  [ 0, 0, 1, 1, 1 ] ],[128X[104X
    [4X[28X[ [ 1, 0, 1, 0, 1 ],[128X[104X
    [4X[28X  [ 0, 1, 0, 0, 0 ],[128X[104X
    [4X[28X  [ -1, -1, 0, 1, 0 ],[128X[104X
    [4X[28X  [ 1, 1, 1, 0, 0 ],[128X[104X
    [4X[28X  [ 0, 0, -1, 0, 0 ] ] ][128X[104X
    [4X[25Xgap>[125X [27XIsAffordingRepresentation( chi, ext );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[28X		    [128X[104X
  [4X[32X[104X
  
  [1X2.3-2 ExtendedRepresentationNormal[101X
  
  [33X[1;0Y[29X[2XExtendedRepresentationNormal[102X( [3Xchi[103X, [3Xrep[103X ) [32X function[133X
  
  [33X[0;0YSuppose  [22XH[122X  is  a  normal  subgroup  of  a group [22XG[122X and [3Xchi[103X is an irreducible
  character  of  [22XG[122X  such  that  the  restriction  of  [3Xchi[103X  to  [22XH[122X,  [22Xphi[122X say, is
  irreducible. If [3Xrep[103X is an irreducible representation of [22XH[122X affording [22Xphi[122X then
  [10XExtendedRepresentationNormal[110X  extends  the  representation  [3Xrep[103X  of  [22XH[122X  to a
  representation  of  [22XG[122X  affording  [3Xchi[103X.  This function is more efficient than
  [10XExtendedRepresentation[110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XG := GL(2,7);;[127X[104X
    [4X[25Xgap>[125X [27Xchi := Irr( G )[ 29 ];;[127X[104X
    [4X[25Xgap>[125X [27XH := SL(2,7);;[127X[104X
    [4X[25Xgap>[125X [27Xphi := RestrictedClassFunction( chi, H );;[127X[104X
    [4X[25Xgap>[125X [27XIsIrreducibleCharacter( phi );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xrep := IrreducibleAffordingRepresentation( phi );;[127X[104X
    [4X[25Xgap>[125X [27Xext := ExtendedRepresentationNormal( chi, rep );[127X[104X
    [4X[28X#I  Need to extend a representation of degree 7. This may take a while.[128X[104X
    [4X[28XCompositionMapping( [(8,15,22,29,36,43)(9,16,23,30,37,44)[128X[104X
    [4X[28X (10,17,24,31,38,45)(11,18,25,32,39,46)(12,19,26,33,40,47)[128X[104X
    [4X[28X (13,20,27,34,41,48)(14,21,28,35,42,49),(2,29,12)(3,36,20)[128X[104X
    [4X[28X (4,43,28)(5,8,30)(6,15,38)(7,22,46)(9,44,14)(10,16,17)[128X[104X
    [4X[28X (11,37,27)(13,23,39)(18,24,25)(19,45,35)(21,31,47)[128X[104X
    [4X[28X (26,32,33)(34,40,41)(42,48,49) ] ->[128X[104X
    [4X[28X[ [ [ -1, 0, 0, 1, 0, -1, 0 ], [ -1, 0, 0, 0, 0, 0, 0 ],[128X[104X
    [4X[28X  [ -1, 1, 0, 0, -1, 0, 0 ], [ 0, -1, 0, 0, 0, 0, 0 ],[128X[104X
    [4X[28X  [ -1, -1, 1, 0, 1, -1, 0 ], [ 0, 0, 0, -1, 0, 0, 0 ],[128X[104X
    [4X[28X  [ -1, 0, 1, -1, 1, 0, -1 ] ],[128X[104X
    [4X[28X  [ [ 1, -1, 0, 1, 0, -1, 1 ], [ 1, 0, -1, 1, -1, 0, 1 ],[128X[104X
    [4X[28X  [ 1, -1, 0, 1, -1, 0, 1 ], [ 0, 0, -1, 0, 0, 0, 0 ],[128X[104X
    [4X[28X  [ -1, 0, 0, 1, 0, -1, 0 ], [ -1, 0, 0, 0, 0, 0, 0 ],[128X[104X
    [4X[28X  [ -1, 1, 0, 0, -1, 0, 0 ] ] ], (action isomorphism) )	[128X[104X
    [4X[25Xgap>[125X [27XIsAffordingRepresentation( chi, ext );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[28X	  [128X[104X
  [4X[32X[104X
  
  
  [1X2.4 [33X[0;0YCharacter Subgroups[133X[101X
  
  [33X[0;0YIf  [22Xχ[122X is an irreducible character of a group [22XG[122X and [22XH[122X is a subgroup of [22XG[122X such
  that  the  restriction  of [22Xχ[122X to [22XH[122X has a linear constituent with multiplicity
  one, then we call [22XH[122X a character subgroup relative to [22Xχ[122X or a [22Xχ[122X-subgroup.[133X
  
  [1X2.4-1 CharacterSubgroupRepresentation[101X
  
  [33X[1;0Y[29X[2XCharacterSubgroupRepresentation[102X( [3Xchi[103X ) [32X function[133X
  [33X[1;0Y[29X[2XCharacterSubgroupRepresentation[102X( [3Xchi[103X, [3XH[103X ) [32X function[133X
  
  [33X[0;0Yreturns  a  representation affording [3Xchi[103X by finding a [3Xchi[103X-subgroup and using
  the  method  described  in [Dix93]. If the second argument is a [3Xchi[103X-subgroup
  then  it  returns  a  representation  affording  [3Xchi[103X without searching for a
  [3Xchi[103X-subgroup. In this case an error is signalled if no [3Xchi[103X-subgroup exists.[133X
  
  [1X2.4-2 IsCharacterSubgroup[101X
  
  [33X[1;0Y[29X[2XIsCharacterSubgroup[102X( [3Xchi[103X, [3XH[103X ) [32X function[133X
  
  [33X[0;0Yis [10Xtrue[110X if [3XH[103X is a [3Xchi[103X-subgroup and [10Xfalse[110X otherwise.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XG := AlternatingGroup( 8 );;[127X[104X
    [4X[25Xgap>[125X [27Xchi := Irr( G )[ 2 ];;[127X[104X
    [4X[25Xgap>[125X [27XH := AlternatingGroup( 3 );;[127X[104X
    [4X[25Xgap>[125X [27XIsCharacterSubgroup( chi, H );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xrep := CharacterSubgroupRepresentation( chi, H );[127X[104X
    [4X[28X[ (1,2,3,4,5,6,7), (6,7,8) ] -> [ [ [[128X[104X
    [4X[28X 1/3*E(3)+2/3*E(3)^2, 0, 0, -E(3), 0, -1/3*E(3)-2/3*E(3)^2, 1 ],[128X[104X
    [4X[28X   [ 2/3*E(3)+4/3*E(3)^2, 0, 1, 0, 0, 1/3*E(3)-1/3*E(3)^2, 0 ],[128X[104X
    [4X[28X   [ 2/3*E(3)+4/3*E(3)^2, 0, 0, 1, 0, 1/3*E(3)-1/3*E(3)^2, 0 ],[128X[104X
    [4X[28X   [ E(3)^2, 0, 0, 0, 0, 0, 0 ],[128X[104X
    [4X[28X   [ 2/3*E(3)+4/3*E(3)^2, 0, 0, 0, 1, 1/3*E(3)-1/3*E(3)^2, 0 ],[128X[104X
    [4X[28X   [ -2/3*E(3)-1/3*E(3)^2, 0, 0, -1, 0, 2/3*E(3)+1/3*E(3)^2, E(3)^2 ],[128X[104X
    [4X[28X   [ 0, 1, 0, 0, 0, 0, 0 ] ],[128X[104X
    [4X[28X [ [ 1, 0, 0, 0, 0, 0, 0 ], [ 0, 1, 0, 0, 0, 0, 0 ],[128X[104X
    [4X[28X   [ 0, 0, 0, 1, 0, 0, 0 ], [ 0, 0, 0, 0, 1, 0, 0 ],[128X[104X
    [4X[28X   [ 0, 0, 1, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 0, 1, 0 ],[128X[104X
    [4X[28X   [ 0, 0, 0, -E(3), E(3), 0, 1 ] ] ][128X[104X
    [4X[28X	[128X[104X
  [4X[32X[104X
  
  [1X2.4-3 AllCharacterPSubgroups[101X
  
  [33X[1;0Y[29X[2XAllCharacterPSubgroups[102X( [3XG[103X, [3Xchi[103X ) [32X function[133X
  
  [33X[0;0Yreturns  a  list  of  all  [22Xp[122X-subgroups  of  [3XG[103X  which  are [3Xchi[103X-subgroups. The
  subgroups are chosen up to conjugacy in [3XG[103X.[133X
  
  [1X2.4-4 AllCharacterStandardSubgroups[101X
  
  [33X[1;0Y[29X[2XAllCharacterStandardSubgroups[102X( [3XG[103X, [3Xchi[103X ) [32X function[133X
  
  [33X[0;0Yreturns   a  list  containing  well  described  subgroups  of  [3XG[103X  which  are
  [3Xchi[103X-subgroups.  This  list  may  contain  Sylow  subgroups and their derived
  subgroups, normalizers and centralizers in [3XG[103X.[133X
  
  [1X2.4-5 AllCharacterSubgroups[101X
  
  [33X[1;0Y[29X[2XAllCharacterSubgroups[102X( [3XG[103X, [3Xchi[103X ) [32X function[133X
  
  [33X[0;0Yreturns  a  list  of  all [3Xchi[103X-subgroups of [3XG[103X among the lattice of subgroups.
  This  function  call  can  be quite expensive for larger groups. The call is
  expensive  in  particular  if the lattice of subgroups of the given group is
  not yet known.[133X
  
  
  [1X2.5 [33X[0;0YEquivalent Representation[133X[101X
  
  [1X2.5-1 EquivalentRepresentation[101X
  
  [33X[1;0Y[29X[2XEquivalentRepresentation[102X( [3Xrep[103X ) [32X function[133X
  
  [33X[0;0Ycomputes  an  equivalent representation to an irreducible representation [3Xrep[103X
  by  transforming  [3Xrep[103X to a new basis by spinning up one vector (i.e. getting
  the  other  basis  vectors  as images under the first one under words in the
  generators).   If   the   input   representation,  [3Xrep[103X,  is  reducible  then
  [10XEquivalentRepresentation[110X  does  not  return  any  mapping.  In this case see
  section 3.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XG := SymmetricGroup( 7 );;[127X[104X
    [4X[25Xgap>[125X [27Xchi := Irr( G )[ 2 ];;[127X[104X
    [4X[25Xgap>[125X [27Xrep := CharacterSubgroupRepresentation( chi );;[127X[104X
    [4X[25Xgap>[125X [27Xequ := EquivalentRepresentation( rep );[127X[104X
    [4X[28X[ (1,2,3,4,5,6,7), (1,2) ] ->[128X[104X
    [4X[28X[ [ [ 0, 0, 0, E(5)+E(5)^2+E(5)^3+2*E(5)^4, -1, -E(5)-E(5)^2-E(5)^3-2*E(5)^4 ],[128X[104X
    [4X[28X   [ E(5)^3-E(5)^4, E(5)^2+E(5)^3+E(5)^4, E(5)+E(5)^3-E(5)^4, -E(5)+E(5)^2[128X[104X
    [4X[28X          -3*E(5)^3-E(5)^4, -E(5)-E(5)^3+E(5)^4, 2*E(5)-2*E(5)^2+2*E(5)^3 ][128X[104X
    [4X[28X    , [ 0, 0, 0, 1, 0, 0 ],[128X[104X
    [4X[28X   [ 0, 4/5*E(5)+3/5*E(5)^2+2/5*E(5)^3+1/5*E(5)^4, E(5), 1, -E(5),[128X[104X
    [4X[28X       6/5*E(5)+2/5*E(5)^2+3/5*E(5)^3+4/5*E(5)^4 ], [ 0, 1, 0, 0, 0, 0 ],[128X[104X
    [4X[28X   [ 0, 0, E(5), 1, -E(5), 2*E(5)+E(5)^2+E(5)^3+E(5)^4 ] ],[128X[104X
    [4X[28X [ [ -1, 0, E(5)+E(5)^2+E(5)^3+2*E(5)^4, -E(5)-E(5)^2-3*E(5)^4,[128X[104X
    [4X[28X    -E(5)-E(5)^2-E(5)^3-2*E(5)^4, E(5)+E(5)^2+3*E(5)^4 ],[128X[104X
    [4X[28X  [ 0, -1, 0, 0, 0, 0 ],[128X[104X
    [4X[28X  [ 0, 0, 0, E(5)+E(5)^2+E(5)^3+2*E(5)^4, -1, -E(5)-E(5)^2-E(5)^3-2*E(5)^4[128X[104X
    [4X[28X     ], [ 0, 0, -1, -E(5)^4, 1, E(5)+E(5)^2+E(5)^3+2*E(5)^4 ],[128X[104X
    [4X[28X  [ 0, 0, -E(5)^4, -E(5)^3+E(5)^4, E(5)+E(5)^2+E(5)^3+2*E(5)^4,[128X[104X
    [4X[28X      E(5)^3-E(5)^4 ], [ 0, 0, 0, 0, 0, -1 ] ] ][128X[104X
    [4X[25Xgap>[125X [27XIsAffordingRepresentation( chi, equ );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[28X	[128X[104X
  [4X[32X[104X
  
