# Coxeter presentation of Hyperoctahedral group $(\mathbb{Z}/2\mathbb{Z})^n \rtimes S_n$.

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?

• You mean a proof if your representation is correct? – draks ... Feb 2 '16 at 12:07
• yes.. this is the coxeter representation which can be obtained from the coxeter graph. I need an explicit proof for this without depending anything on root systems and simple systems. – Anupam Ah Feb 5 '16 at 9:16
• What homomorphism defines the semidirect product? – Shaun Nov 29 '18 at 22:57