Matrix Representation of Wreath Product My aim is to understand the wreath product of $\mathbb{Z}/2$ by $S_3$ in terms of group of matrices. The problem I am facing is the following:
(1) In GAP, the wreath product of $\mathbb{Z}/2$ by $S_3$ is understood as: take $|S_3|=6$ copies of $\mathbb{Z}/2$, and consider their direct product. On this direct product, $S_3$ acts as a regular permutation group of degree $6$. The resulting group $\mathbb{Z}/2\wr S_3$ has order $2^6.|S_3|=2^7.3$. 
(2) There is a mathoverflow question with similar title. In the answer to this question, G. Robinson understood the  wreath product of $\mathbb{Z}/2$ by $S_3$ as: consider elementary abelian $2$-group of rank $3$, and let $S_3$ act on it by permutation group of degree $3$. The resulting semidirect product of $(\mathbb{Z}/2)^3$ by $S_3$ is the wreath product $\mathbb{Z}/2 \wr S_3$ (order of this group is $2^3.6=48$).
The wreath product considered by G. Robinson has nice interpretation as a group of integer matrices. But, how can we describe nicely the wreath product 
$\mathbb{Z}/2\wr S_3$ (as considered in GAP) as a group of integer matrices.
 A: You are asking about the monomial representation of the wreath product of permutation group acting on copies of a cyclic group.
If $G$ is a permutation group on $m$ points, and $C$ is a cyclic group of order $n$, then $C \wr G \cong C^m \rtimes G$ can be considered a matrix group $MW$ by considering $C$ as a group of $1\times 1$ matrices (the $n$th roots of unity) and then $MW = \{ a(g,c_1,\ldots,c_m) : g \in G, c_i \in C \}$ where the $(i,j)$th entry of $a(g,c_1,\ldots,c_m)$ is 0 unless $g(i) = j$ and is $c_i$ if $g(i) = j$.
In other words, the permutation is a permutation matrix describing where the nonzero entries are, and the $c_i$ describe what the nonzero entries are.
This can be done in GAP using the following commands:
`
gap> G := SymmetricGroup(3);;
gap> C := CyclicGroup(IsPermGroup,2);;
gap> W := WreathProduct(C,G);;
gap> MG := Group( List( GeneratorsOfGroup( G ), perm ->
> PermutationMat( perm, NrMovedPoints(G) ) ) );;
gap> MC := Group( DiagonalMat( Concatenation( [ E(Size(C)) ],
> List( [2..NrMovedPoints(G)], i -> 1 ) ) ) );;
gap> MW := ClosureGroup( MG, MC );;
`
If you need more general $C$ (not just cyclic), then you need to use block monomial matrices. InducedGModule will do most of the work for you.
In order to make $W$ a permutation group (so that faster methods are available), you'll want to pass it permutation groups as arguments. CyclicGroup(IsPermGroup,2) does this, but so does SymmetricGroup(2).
If you really want the group of order $2^6 \cdot 6$, then change the first line to the following:
`
gap> G := Image(RegularActionHomomorphism(SymmetricGroup(3)));;
`
This writes $S_3$ as a permutation group on 6 points.
