 Permutation covering of a wedge of circles

Choose permutations ${\alpha }_1,\dots ,{\alpha }_r\in S_n$. A covering of a wedge of $r$ oriented circles has fibre $\{1,\dots ,n\}$ and an edge labelled $j$ from $i$ to ${\alpha }_j(i)$ above the $j$th circle. It is connected exactly when the generated permutation group is transitive, and its deck group is the centralizer of that group in $S_n$.