Structure of Clifford groups of composite finite quantum systems

In this paper the Clifford groups of general multipartite quantum systems with configuration space $\mathbb{Z}_{n_1}\oplus\cdots\oplus\mathbb{Z}_{n_k}$, $k\geq 1$ are studied. It is known that the Clifford group is a natural semidirect product provided the dimension $N=n_1\cdots n_k$ of the corresponding Hilbert space is an odd number. For even $N$ special results on the Clifford groups are scattered in the mathematical literature, but they do not concern the semidirect product structure. Using the relation of generators of the associated symplectic group $\mathrm{Sp}_{[n_1,\dots,n_k]}$ we prove that for even $N=n_1\cdots n_k$ both Clifford group and the projective Clifford group are natural semidirect products if and only if $N$ is not divisible by four.