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

[Alexander Konovalov]

P.S. This computation now works too in GAP 4.6.3 and Image(phi2,b)^2
returns an identity matrix. Thanks for reporting this bug!

Best wishes,
Alexander


On 10 Dec 2012, at 22:04, Neha Wadhwani <nehamakhijani at gmail.com> wrote:

> 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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