Any $f\in Möb(\mathbb{C})$ can be represented as a composition of less than $6$ transformations of the following form:

1. (Translation) 1;; $z\mapsto z+b\sim \left[\begin{array}{cc}1& b\\ 0& 1\end{array}\right]$.
2. (Dialation/rotation) 1;;$z\mapsto az\sim \left[\begin{array}{cc}a& 0\\ 0& 1\end{array}\right]$
3. (Inversion) 1;;$z\to \frac{1}{z}\sim \left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]$