The Monodromy group of $pq$-covers

In this work we study the monodromy group of covers $\varphi \circ \psi$ of curves \linebreak $\mathcal{Y}\xrightarrow {\quad {\psi}} \mathcal{X} \xrightarrow {\quad \varphi} \mathbb{P}^{1}$, where $\psi$ is a $q$-fold cyclic \'etale cover and $\varphi$ is a totally ramified $p$-fold cover, with $p$ and $q$ different prime numbers with $p$ odd. We show that the Galois group $\mathcal{G}$ of the Galois closure $\mathcal{Z}$ of $\varphi \circ \psi$ is of the form $\mathcal{G} = \mathbb{Z}_q^s \rtimes \mathcal{U}$, where $0 \leq s \leq p-1$ and $\mathcal{U}$ is a simple transitive permutation group of degree $p$. Since the simple transitive permutation group of prime degree $p$ are known, and we construct examples of such covers with these Galois groups, the result is very different from the previously known case when the cover $\varphi$ was assumed to be cyclic, in which case the Galois group is of the form $\mathcal{G} = \mathbb{Z}_q^s \rtimes \mathbb{Z}_p$. Furthermore, we are able to characterize the subgroups $\mathcal{H}$ and $\mathcal{N}$ of $\mathcal{G}$ such that $\mathcal{Y} = \mathcal{Z}/\mathcal{N}$ and $X = \mathcal{Z}/\mathcal{H}$.