  We study several families of semisimple Hopf algebras, arising as bismash
products, which are constructed from finite groups with a certain specified
factorization. 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 a
comultiplication twist also have this property and are thus non-trivial. We
also show that bismash products constructed from Frobenius groups do have the
structure (as algebras) of group algebras.
