Let $X$, $Y$ be smooth, connected, compact manifolds (for instance, projective varieties) and $f \colon X \longrightarrow Y$ be a finite, branched cover of degree $n$, with branch locus $B \subset Y$. We can then associate to $f$ its monodromy representation $$\theta_f \, \colon \pi_1(Y-B) \longrightarrow S_n,$$ so that $\mathrm{im} \, \theta_f$ is a transitive subgroup of $S_n$. Conversely, isomorphism classes of connected covers of $Y$, branched over $B$, are in bijection to monodromy representations of the type above, up to conjugacy in $S_n$.

We now assume that $f$ is *simple*, that is that for all $b \in B$ the fibre $f^{-1}(b)$ consists of exactly $n-1$ points, and we call $f$ an *elementary* cover if it is not possible to factor it as $$X \stackrel{g}{\longrightarrow} Z \stackrel{h}{\longrightarrow} Y,$$
where $g\colon X \longrightarrow Z$ is a branched cover and $h \colon Z \longrightarrow Y$ is an ordinary (i.e, unramified) cover.

Q.What is the characterization of simple, elementary covers $f \colon X \longrightarrow Y$, branched over $B$, in terms of their monodromy representation $\theta_f$?