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We study several families of semisimple Hopf algebras, arising as bismashproducts, which are constructed from finite groups with a certain specifiedfactorization. First we associate a bismash product $H_q$ of dimension$q(q-1)(q+1)$ to each of the finite groups $PGL_2(q)$ and show that these $H_q$do not have the structure (as algebras) of group algebras (except when $q=2,3$). As a corollary, all Hopf algebras constructed from them by acomultiplication twist also have this property and are thus non-trivial. Wealso show that bismash products constructed from Frobenius groups do have thestructure (as algebras) of group algebras.

On the algebra structure of some bismash products