Recently I'm working on Exercise I of Problem III.3 of "Algebraic curves and Riemann surfaces".
Define holomorphic and effective actions of $A_4,S_4$ and $A_5$ on the projective line such that the quotient map has $3$ branched points with ramification numbers $(2,3,3)$, $(2,3,4)$ and $(2,3,5)$ respectively.
But I have no idea about how these groups act on the projective line, could anyone give me some hints or help? Thanks in advance.
The groups are the isometry groups of a tetrahedron, a cube, and a dodecahedron respectively. They are contained in $SO(3)$ which operates isometrically on the sphere i.e. the complex projective line. ($SO(3)$ is also a maximal compact subgroup of $PSL(2,C)$. See https://groupprops.subwiki.org/wiki/Classification_of_finite_subgroups_of_SO(3,R)