Aschbacher Class $2$ subgroup structure

Question

In $PGL(12,3)$, there should be an Aschbacher Class $2$ subgroup the image of $GL(2,3)^6 \wr{\rm Sym}(6)$. I am trying to locate the image of $GL(2,3)^6$ in Magma using derived subgroup but it doesn't seem to be doing the job.

     X:=GL(12,3);
     G:=PGL(12,3);
     ro:=hom<X->G|G.1,G.2>;
     d:=ClassicalMaximals("L",12,3:general:=true,classes:={2});
     #ro(d[4]);
    
     M4:=ro(d[4]);
     DER:=DerivedSubgroup(M4);
     #DER;
     1100753141760
     DER1:=DerivedSubgroup(DER);
     #DER1;
     1100753141760

Which part went wrong? How do I locate the desired part then? Thank you.

Answer 1

You want the kernel of the imprimitive action in the group $\mathtt{d[4]}$.

> d:=ClassicalMaximals("L",12,3:general:=true,classes:={2});
> H := d[4]; ng := Ngens(H); ng;
4
> IsPrimitive(H);
false
> rho := hom< H->Sym(6) | [ImprimitiveAction(H,H.i) : i in [1..4]] >;
> K := Kernel(rho);
> isiso := IsIsomorphic(K, DirectProduct([GL(2,3): i in [1..6]])); isiso;
true