In this paper, we construct a symmetric group ${ \rm Sym}_{ 2(4^n-1)}$, which contains a subgroup isomorphic to the $n$-qubit projective Clifford group $\mathcal{ C}_n$. To establish this result, we investigate the centralizers of the $z$ gate and the phase gate within the $n$-qubit projective Clifford group, utilizing the normal form of the Clifford operators. As a byproduct, we also provide a presentation of the inertia subgroup of $\mathcal{ C}_n$.
Categories: math.GR