Let $G$ be a finite group acting on the Riemann sphere. Using Hurwitz' formula, it can be easily seen that the quotient map $\pi: G \to $ there could be two or three branch points. The case of two branches gives rise to a cyclic group. If we have three branches, four cases occur. One of them is the case $|G|=12$ (order of $G$ is $12$). Miranda (Algebraic Curves and Riemann Surfaces) says that in this case $G=A_4$. So, my question is: how is it possible to define an action of $A_4$ on the Riemann sphere?
Using the presentation $\langle r,t: \, r^3=t^3=(rt)^2=1 \rangle$, I have to find two maps $r,t$ on $\mathbb{C}_\infty$ having order $3$ and whose composition has order $2$. I have no clue.