On the irreducibility of the space of Galois covers of curves of a given numerical type for certain split metacyclic groups

We consider the spaces Mg(G)⊂Mg of curves of genus g≥2, that admit an effective action by a given finite group G and want to understand its irreducible components. We prove that if G=Cm⋊Cn is a split metacyclic group where both factors have odd prime order, the irreducible components are determined by a numerical datum which comes from the monodromy of the induced G-coverings.