In this paper we study the Galois group of cover composition a q-cyclic étale and cyclic p-gonal for any odd prime p. Furthermore, give properties isogenous decompositions certain Prym Jacobian varieties associated to intermediate subcovers given by subgroups.

To every double cover ramified in two points of a general trigonal curve genus g, one can associate an etale tetragonal g+1. We show that the corresponding Prym varieties are canonically isomorphic as principally polarized abelian varieties.