On group theoretical Hopf algebras and exact factorization of finite groups

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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 Domega(Sigma), for some finite group Sigma and some 3-cocycle omega on Sigma. We show that semisimple Hopf algebras obtained as bicrossed products from an exact factorization of a finite group Sigma are group theoretical. We also describe their Drinfeld double as a twisting of Domega(Sigma), for an appropriate 3-cocycle omega coming from the Kac exact sequence.