Suborbital graphs of a extended congruence subgroup by Fricke involution

Let p be a fixed prime and let Gamma(0) (p) denote the usual subgroup of Gamma = PSL2(Z) = SL2(Z)/{+/- I}, consisting of all the matrices with lower left entry divisible by p. Then the attached Fricke group is given by Gamma(0)(p) boolean OR Gamma(0)(p)W-p, W-p := 1/root p{(0)(1)(P)(0)). The Fricke group acts on the upper half-plane. Its action on Q boolean OR {infinity} is transitive but imprimitive. We study the action of Fricke group on the projective line Q boolean OR {infinity} by using suborbital graphs. These are directed graphs with vertex-set Q boolean OR {infinity}, their edge-sets being the orbits of the group on the cartesian square [Q boolean OR {infinity}](2).