  We give a path integral construction of the quantum mechanical partition
function for gauged finite groups. Our construction gives the quantization of a
system of $d$, $N\times N$ matrices invariant under the adjoint action of the
symmetric group $S_N$. The approach is general to any discrete group. For a
system of harmonic oscillators, i.e. for the non-interacting case, the
partition function is given by the Molien-Weyl formula times the zero-point
energy contribution. We further generalise the result to a system of non-square
and complex matrices transforming under arbitrary representations of the gauge
group.
