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In this paper we study the Galois group of the Galois cover of thecomposition of a $q$-cyclic \'etale cover and a cyclic $p$-gonal cover for anyodd prime $p$. Furthermore, we give properties of isogenous decompositions ofcertain Prym and Jacobian varieties associated to intermediate subcovers givenby subgroups.

$q$-\'etale covers of cyclic $p$-gonal covers