$f: (\Sigma; p_1, \ldots, p_n) \to (\mathbb{CP}^1; \infty)$ such that
ramification can only occur over $\infty$ or the $m$th roots of unity; and $f^{-1}(\infty) = \mu_1 p_1 + \cdots + \mu_n p_n$.
An automorphism of a branched cover $f: \Sigma \to \mathbb{CP}^1$ is an automorphism $g: \Sigma \to \Sigma$ such that $f \circ g = f$.
In particular, I want to count |Aut(f)| where $f$ is a degree 3 map from a genus 1 surface to $\mathbb{CP}^1$ and $f^{-1}(\infty)=3p_1$ and $m=2$ that there are two ramification point other than infinity.
Let me also specify the ramification profile over those two points. We want the ramification to be 3 cycle. Hence all the point have same ramification profile.
I heard the Aut(f) should be a subgroup of $S_3$. I am not really clear about the concept. How would we count it?
Is the count 3?