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!*
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