On character table of Clifford groups

Based on a presentation of Cn and the help of [The GAP Group 2022, https://www.gap-system.org] we construct the character table of the Clifford group Cn 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 Helsen et al. [J. Math. Phys. 59, 072201 (2018)] and Wallman and Flammia [New J. Phys. 16(10), 103032 (2014)] and reveal some new phenomena. We prove that when n ≥ 3, (1) the trivial character is the only linear character for Cn and hence Cn equals to its commutator subgroup, (2) the n-qubit Pauli group Pn is the only proper non-trivial normal subgroup of Cn, (3) the matrix representation M2n is a faithful representation for Cn. As a byproduct, we give a presentation of the finite symplectic group Sp(2n, 2) in terms of generators and relations.