On character table of Clifford groups

Based on a presentation of ${\mathcal{C}}_n$ and the help of [GAP], we construct the character table of the Clifford group ${\mathcal{C}}_n$ for $n=1,2,3$. As an application, we can efficiently decompose the (higher power of) tensor product of the matrix representation in those cases. Our results recover some known results in [HWW, WF] and reveal some new phenomena. We prove that when $n\ge 3$, (1) the trivial character is the only linear character for ${\mathcal{C}}_n$ and hence ${\mathcal{C}}_n$ equals to its commutator subgroup, (2) the $n$-qubit Pauli group ${\mathcal{P}}_n$ is the only proper non-trivial normal subgroup of ${\mathcal{C}}_n$, (3) the matrix representation ${\mathcal{M}}_{2^n}$ is a faithful representation for ${\mathcal{C}}_n$. As a byproduct, we give a presentation of the finite symplectic group $Sp(2n,2)$ in terms of generators and relations.