Defining relations of a group Γ=G3,4(2,Z) and its action on real quadratic field

In this paper‎, ‎we have shown that the coset diagrams for the‎ ‎action of a linear-fractional group Γ generated by the linear-fractional‎ ‎transformations r:z→z−1/z and s:z→−1/2(z+1) on‎ ‎the rational projective line is connected and transitive‎. ‎By using coset diagrams‎, ‎we have shown that r³=s^4=1 are defining relations for Γ‎. ‎Furthermore‎, ‎we have studied some important results for the action of group Γ on real‎ ‎quadratic field Q(√n)‎. ‎Also‎, ‎we have classified all the ambiguous numbers in the orbit.