  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 $Z_6Z_6$ 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.
