Generators and relations for the group $U_4(\mathbb{Z}[1\mathrm{/}\sqrt{2},i])$

We give a presentation by generators and relations of the group $U_4(\mathbb{Z}[1\mathrm{/}\sqrt{2},i])$ of unitary $4\times 4$ matrices with entries in the ring $\mathbb{Z}[1\mathrm{/}\sqrt{2},i]$. This is motivated by the problem of exact synthesis for the Clifford+T gate set in quantum computation.