[GAP Forum] Automorphism Group as Permutation group on Conj.Classes

[Alexander Hulpke]

Dear GAP-forum, dear Giulio Peruginelli,

> I have a finite group G and I want to compute a representation of  
> Aut(G)
> as a permutation group on the set of the conjugacy classes of G.

>
> So I wrote down these lines in Gap 3

>    AutG:=AutomorphismGroup(G);
>    C:=ConjugacyClasses(G);
>    D:=[];
>    for k in [1..Length(C)] do
>       D[k]:=Representative(C[k]);
>    od;
>    AutG1:=Operation(AutG,D);

This ``action'' on the representatives of classes is not well defined  
-- the image of a representative is not necessarily a representative.  
Instead you will have to act on the classes themselves. For this you  
will have to define your own action. The following is GAP4 code for it:
(In GAP3 you would need `cl.group' instead of `ActingDomain(cl)' and  
`Operation' instead of `Action' I believe.)

OnClasses:=function(cl,g) # function to describe the action
   return ConjugacyClass(ActingDomain(cl),Representative(cl)^g);
end;

AutG:=AutomorphismGroup(G);
C:=ConjugacyClasses(G);
Action(AutG,C,OnClasses);

If you need the connection to the original group you might prefer:
ActionHomomorphism(AutG,C,OnClasses,"surjective");

instead of `Action'.

Best,

     Alexander Hulpke


-- Colorado State University, Department of Mathematics,
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