Wrong Generators for Matrix Wreath Product

[FriedrichRober]

Edit: The observed issue arises, since the generators of the group are set wrong. It has nothing to do with ConjugacyClasses. See comments below for further details.

Observed behaviour

gap> K := GL(2,3);;
gap> H := Group((1,2,3)(4,5));;
gap> G := WreathProduct(K, H);
<matrix group of size 1528823808 with 3 generators>
gap> Size(ConjugacyClasses(G));
384

Expected behaviour

gap> Size(ConjugacyClasses(G));
8448

The number 8448 is tested with Magma and my package WreathProductElements (which use different approaches to obtain this result)

Magma Code:

> K := GL(2,3);
> S := SymmetricGroup(5);      
> H := sub<S | S!(1,2,3)(4,5)>;
> G := WreathProduct(K, H);
> #G;
1528823808
> #ConjugacyClasses(G);    
8448

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[FriedrichRober]

Of corse it could be that both Magma and my package obtain a wrong a result. I will further investigate this issue.

[FriedrichRober]

Ok, I found the issue. The group generators are wrong:

gap> K := GL(2,3);;
gap> H := Group((1,2,3)(4,5));;
gap> G := WreathProduct(K, H);;
gap> Size(GeneratorsOfGroup(G));
3
gap> Display(G.1);
 2 . . . . . . . . .
 . 1 . . . . . . . .
 . . 1 . . . . . . .
 . . . 1 . . . . . .
 . . . . 1 . . . . .
 . . . . . 1 . . . .
 . . . . . . 1 . . .
 . . . . . . . 1 . .
 . . . . . . . . 1 .
 . . . . . . . . . 1
gap> Display(G.2);
 2 1 . . . . . . . .
 2 . . . . . . . . .
 . . 1 . . . . . . .
 . . . 1 . . . . . .
 . . . . 1 . . . . .
 . . . . . 1 . . . .
 . . . . . . 1 . . .
 . . . . . . . 1 . .
 . . . . . . . . 1 .
 . . . . . . . . . 1
gap> Display(G.3);
 . . 1 . . . . . . .
 . . . 1 . . . . . .
 . . . . 1 . . . . .
 . . . . . 1 . . . .
 1 . . . . . . . . .
 . 1 . . . . . . . .
 . . . . . . . . 1 .
 . . . . . . . . . 1
 . . . . . . 1 . . .
 . . . . . . . 1 . .

But since the top group H is not transitive on [1 .. 5], we need further generators in the base group.

[FriedrichRober]

I will try to fix this issue, since I am already familiar with the WreathProduct code in GAP