  The Clifford group is a fundamental structure in quantum information with a
wide variety of applications. We discuss the tensor representations of the
$q$-qubit Clifford group, which is defined as the normalizer of the $q$-qubit
Pauli group in $U(2^q)$. In particular, we characterize all irreducible
subrepresentations of the two-copy representation $\varphi^{\otimes2}$ of the
Clifford group on the matrix space $\mathbb{C}^{d\times d}\otimes
\mathbb{C}^{d\times d}$ with $d=2^q$. In an upcoming companion paper we applied
this result to cut down the number of samples necessary to perform randomised
benchmarking, a method for characterising quantum systems.
