Constructing $\mathrm{SL}(2,5)\rtimes\mathbb{Z}_{11}^2$ in GAP

abstract-algebragapsemidirect-product

I am trying to construct the semidirect product $\mathrm{SL}(2,5)\rtimes\mathbb{Z}_{11}^2$ in GAP, where $\mathrm{SL}(2,5)$ is the subgroup $\left\langle\begin{pmatrix}4&1\\0&3\end{pmatrix},\begin{pmatrix}0&3\\7&10\end{pmatrix}\right\rangle\leq\mathrm{SL}(2,11)$ (which is isomorphic to $\mathrm{SL}(2,5)$) and acts on $\mathbb{Z}_{11}^2$ via matrix-vector-multiplication.
This subgroup I already implemented:

gap> x:=[[Z(11)^2,Z(11)^0],[0*Z(11),Z(11)^8]];
[ [ Z(11)^2, Z(11)^0 ], [ 0*Z(11), Z(11)^8 ] ]
gap> y:=[[0*Z(11),Z(11)^8],[Z(11)^7,Z(11)^5]];
[ [ 0*Z(11), Z(11)^8 ], [ Z(11)^7, Z(11)^5 ] ]
gap> G:=Group(x,y);
Group([ [ [ Z(11)^2, Z(11)^0 ], [ 0*Z(11), Z(11)^8 ] ], [ [ 0*Z(11), Z(11)^8 ], [ Z(11)^7, Z(11)^5 ] ] ])
gap> Order(G);
120
gap> IsSL(G);
true

However I am not sure how to continue from here. I tried writing $\mathbb{Z}_{11}$ as 

gap> ElementaryAbelianGroup(121)

but I couldn't make it so that I can multiply a matrix with a "vector".
Then I tried to write it as

gap> V:=VectorSpace(GF(11),[[Z(11),0*Z(11)],[0*Z(11),Z(11)]]);
<vector space over GF(11), with 2 generators>

which seems more promising. But when trying to define the semidirect product, I didn't know how to proceed. How do I specify the operation? I should add that I haven't been using GAP for that long, so I've been trying to find the info on GAP I need on the Internet, since examining the structure of this group by hand seemed a little overambitious.

Help would be very much appreciated!

Best Answer

Formally, you would have to create automorphisms of ElementaryAbelianGroup(121) and construct a homomorphism from $G$ into that group. Since this is a lot of work, as a convenience also the following shortcut (slightly abusing notation) works:

gap> S:=SemidirectProduct(G,GF(11)^2);
<matrix group of size 14520 with 3 generators>

(The group is represented in dimension $n+1$ as subgroup of $AGL(n,p)$.)

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