  Let X=GM be a finite group factorisation. It is shown that the quantum double
D(H) of the associated bicrossproduct Hopf algebra $H=kM\cobicross k(G)$ is
itself a bicrossproduct $kX\cobicross k(Y)$ associated to a group YX, where
$Y=G\times M^{op}$. This provides a class of bicrossproduct Hopf algebras which
are quasitriangular. We also construct a subgroup $Y^\theta X^\theta$
associated to every order-reversing automorphism $\theta$ of X. The
corresponding Hopf algebra $kX^\theta\cobicross k(Y^\theta)$ has the same
coalgebra as H. Using related results, we classify the first order bicovariant
differential calculi on H in terms of orbits in a certain quotient space of X.
