Monodromy group of a Belyi map (awaiting review)

The monodromy group of a Belyi map $\phi \colon X \to \mathbb{P}^1$ of degree $d$ is the transitive subgroup of $S_d$ generated by its associated transitive permutation triple $\sigma$. This group is the geometric monodromy group of the branched cover $\phi$, or equivalently it is the geometric Galois group of the corresponding extension of function fields $K(X) \supseteq K(\phi) \simeq K(\mathbb{P}^1)$.