I am looking for a holomorphic action of $C_{2}\times C_{2}$ on $\mathbb{P}^1$. Is it true that there is a unique effective action given by $ a:z\rightarrow -z, \ \ \ \ b:z\mapsto 1/z $ up to change of variable? Here $a$ is the generators of the first (second) factor of $C_{2}\times C_{2}$.
What about $C_{3}\times C_{3}$? I could not find any effective action.