On group theoretical Hopf algebras and exact factorizations of finite groups

We show that a semisimple Hopf algebra A is group theoretical if and only if its Drinfeld double is a twisting of the Dijkgraaf–Pasquier–Roche quasi-Hopf algebra Dω(Σ), for some finite group Σ and some ω Z3(Σ,k×). We show that semisimple Hopf algebras obtained as bicrossed products from an exact factorization of a finite group Σ are group theoretical. We also describe their Drinfeld double as a twisting of Dω(Σ), for an appropriate 3-cocycle ω coming from the Kac exact sequence.