A Note on Transitive Action of the Extended Modular Group on Rational Numbers

The extended modular group $\overline{\Gamma}$ is isomorphic to the amalgamated free product of two dihedral groups $D_2$ and $D_3$ with amalgamation $\mathbb{Z}_2$. This acts on rational numbers transitively. In this study, we obtain elements in that are mappings between given rationals. Also, express these as a word generators. We use interesting relations continued fractions Farey graph.