Is the mapping class group for the twice punctured torus known?
I know that the mapping class group for both the torus and the once punctured torus is $SL_2(\mathbb{Z})$ and that the action on the modular parameter $\tau$ of the torus is $\tau \rightarrow \frac{a\tau+b}{c\tau+d}$ where $\begin{pmatrix}a&b\\c&d\end{pmatrix} \in SL_2(\mathbb{Z})$.
Is a similar expression for the twice punctured torus know? I would like to know how it explicitly acts on a function defined on this surface similar to the above mapping for $\tau$.