dear all, Let $H=Z\_2 \wr Z\_2 \wr...\wr Z\_2$ ( r times), I need to know the structure of H as a matrix group.
Thanks in advance



wreath product and matrix presentation
Since it came up in comments, I will give an answer about the group I believe was intended to be asked about. A monomial matrix is a matrix which has one non-zero entry in each row and one non-zero entry in each column. The monomial $n \times n$ matrices who non-zero entries are all $\pm 1$ form a group, which may be thought of as an abstract group as $Z\_{2} \wr S\_{n}.$ This matrix group has a normal elementary Abelian subgroup of order $2^{n}$ consisting of all its diagonal matrices. This group has many Sylow $2$-subgroups when $n >2,$ but they are all conjugate within it, so in particular are all isomorphic, and they all contain the normal subgroup consisting of all its diagonal matrices. I believe that the group which was intended to be asked about was such a Sylow $2$-subgroup in the case that $n = 2^{r-1}$.


