Classification of differentials and Cartan calculus on bicrossproducts

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We provide the Cartan calculus for bicovariant differential forms on bicrossproduct quantum groups k(M) kG associated to finite group factorizations X=GM and a field k. The irreducible calculi are associated to certain conjugacy classes in X and representations of isotropy groups. We find the full exterior algebras and show that they are inner by a bi-invariant 1-form θ which is a generator in the noncommutative de Rham cohomology H¹. The special cases where one subgroup is normal are analysed. As an application, we study the noncommutative cohomology on the quantum codouble D*(S₃) k(S₃) k₆ and the quantum double D(S₃)=k(S₃) kS₃, finding respectively a natural calculus and a unique calculus with H⁰=k.1.