On the exponent of tensor categories coming from finite groups

We describe the exponent of a group-theoretical fusion category $\mathcal C = \mathcal C(G, \omega, F, \alpha)$ associated to a finite group $G$ in terms of group cohomology. We show that the exponent of $\C$ divides both $e(\omega) \exp G$ and $(\exp G)^2$, where $e(\omega)$ is the cohomological order of the 3-cocycle $\omega$. In particular $\exp \C$ divides $(\dim \C)^2$.