On the exponent of tensor categories coming from finite groups

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