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 geq 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}_2ⁿ 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.