On Suborbital Graphs for the Extended Modular Group {hat{Gamma}}

In this paper, we show that the extended modular group ${hat{Gamma}}$ acts on ${hat{mathbb{Q}}}$ transitively and imprimitively. Then the number of orbits of ${hat{Gamma} _{0}(N)}$ on ${hat{mathbb{Q}}}$ is calculated and compared with the number of orbits of ${Gamma _{0}(N)}$ on ${hat{mathbb{Q}}}$ . Especially, we obtain the graphs ${hat{G}_{u, N}}$ of ${hat{Gamma}_{0}(N)}$ on ${hat{mathbb{Q}}}$ , for each ${Ninmathbb{N}}$ and each unit ${u in U_{N} }$ , then we determine the suborbital graph ${hat{F}_{u,N}}$ . We also give the edge conditions in ${hat{G}_{u, N}}$ and the necessary and sufficient conditions for a circuit to be triangle in ${hat{F}_{u, N}.}$