Triangle groups and PSL2(q)

We consider hyperbolic triangle groups of the form T = T-p1,T-p2,T-p3, where p1, p2, p3 are prime numbers. Let p be a prime number and n be a positive integer. We give a necessary and sufficient condition for L-2(p(n)) to be the image of a given hyperbolic triangle group, where L-2(p(n)) denotes the projective special linear group PSL2(p(n)). It follows that, given a prime number p, there exists a unique positive integer n such that L-2(p(n)) is the image of a given hyperbolic triangle group. Finally, given a hyperbolic triangle group T, we determine the asymptotic probability that a randomly chosen homomorphism phi : T -> L-2(p(n)) is surjective, as p(n) tends to infinity.