I know that the hyperoctahedral group $(\mathbb{Z}/2\mathbb{Z})^n \rtimes S_n$ has the presentation
$$\langle s_{\text{1}},\ldots,s_n\mid s_{\text{1}}^{\text{2}}=s_i^2=1, (s_1s_2)^4=(s_is_{i+1})^3=(s_is_j)^2=1~| ~2\leq i\leq n, i+2\leq j\leq n \rangle,$$
where the generator $s_1$ is the diagonal matrix with a $-1$ in the top left entry and $1$s on the rest of the diagonal entry and $s_i$ are the permutation matrix formed from the identity switching both the $i-1$st and $i$th rows and corresponding columns.
Where can I get a proof for this?