Transitive factorizations in the hyperoctahedral group

The classical Hurwitz enumeration problem has a presentation in terms of transitive factorizations in the symmetric group. This presentation suggests a generalization from type-A to other finite reflection groups and, in particular, to type-B. We study this generalizaztion both from a combinatorial and a geometric point of view, with the prospect of providing a mean of understanding more of the structure of the moduli spaces of maps with an S_2-symmetry. The type-A case has been well studied and connects Hurwitz numbers to the moduli space of curves. We conjecture an analogous setting for the type-B case that is studied here.