Finite Group Factorizations and Braiding

We compute the quantum double, braiding and other canonical Hopf algebra constructions for the bicrossproduct Hopf algebra H associated to the factorization of a finite group into two subgroups. The representations of the quantum double are described by a notion of bicrossed bimodules, generalising the cross modules of Whitehead. We also show that self-duality structures for the bicrossproduct Hopf algebras are in one-one correspondence with factor-reversing group isomorphisms. The example Z6Z6 is given in detail. We show further that the quantum double D(H) is the twisting of D(X) by a non-trivial quantum cocycle, where X is the associated double cross product group.