Embedding the $n$-Qubit Projective Clifford Group into a Symmetric Group

In this paper, we construct a symmetric group ${\mathrm{S}\mathrm{y}\mathrm{m}}_{2(4^n-1)}$, 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 ${\mathcal{C}}_n$.