[GAP Forum] Irreducible representation of dihedral group D20 over GF(8)

[Neha Wadhwani]

Hi

I am not able to understand if the following is actually a well defined
homomorphism!

*G:=DihedralGroup(20);*
<pc group of size 20 with 3 generators>

*b:=G.1*G.2;*
f1*f2

*b^2;*
<identity> of ...

*phi:=IrreducibleRepresentations(G,GF(8));*
[ Pcgs([ f1, f2, f3 ]) -> [ [ [ Z(2)^0 ] ], [ [ Z(2)^0 ] ], [ [ Z(2)^0 ] ]
],
  Pcgs([ f1, f2, f3 ]) ->
    [ [ [ Z(2)^0, 0*Z(2), Z(2^3), Z(2)^0 ], [ 0*Z(2), Z(2)^0, Z(2^3)^6,
               Z(2^3)^3 ], [ 0*Z(2), 0*Z(2), Z(2^3), Z(2^3)^3 ],
          [ 0*Z(2), 0*Z(2), Z(2^3)^2, Z(2)^0 ] ],
      [ [ Z(2^3)^6, Z(2^3)^5, Z(2^3)^6, Z(2^3) ],
          [ Z(2^3)^4, Z(2^3), Z(2)^0, Z(2^3)^5 ],
          [ Z(2^3)^2, Z(2^3)^6, Z(2^3)^3, Z(2^3)^4 ],
          [ Z(2^3)^5, Z(2)^0, Z(2^3)^3, Z(2^3)^6 ] ],
      [ [ Z(2^3)^2, Z(2^3)^4, Z(2^3), 0*Z(2) ],
          [ Z(2^3)^3, Z(2^3), 0*Z(2), Z(2^3) ],
          [ Z(2^3), Z(2^3), Z(2^3)^3, Z(2^3)^5 ],
          [ Z(2)^0, 0*Z(2), Z(2^3)^4, Z(2^3)^2 ] ] ] ]

*phi2:=phi[2];;*
*
*
*Image(phi2,b)^2;*
[ [ Z(2^3)^6, Z(2^3)^3, 0*Z(2), 0*Z(2) ],
  [ Z(2^3)^2, Z(2^3)^4, 0*Z(2), 0*Z(2) ],
  [ Z(2^3)^4, Z(2^3)^3, Z(2^3), Z(2^3)^6 ],
  [ Z(2^3)^2, Z(2^3)^6, Z(2^3)^5, Z(2^3)^5 ] ]
*which is not an identity matrix..*
*
*
*Please let me know if I am going wrong....*
*
*
*Thanks!*