Parametrization of all homomorphisms from PGL(2Z) into PGL{2,q)

Let PL(F q) denote the projective line over a Galois field F q. Consider PSL (2, Z ) as a free product of two cyclic groups and of orders 2 and 3. We have shown that any homomorphism from PSL(2,Z) into PGL(2,q) can be extended to a homomorphism from PGL(2Z) into PGL(2q) except in the case where the order of the image of xyis 6 but the images of xand ydo not commute in PGL(2q). It has been shown also that every element in PGL(2,q), not of order 1,2 , or 6, is the image of xyunder some non-degenerate homomorphism. We have parametrized the conjugacy classes of non-degenerate homomorphisms α with the non-trivial elements of F q. Due to this parametrization we have developed a useful mechanism by which one can construct. a unique coset diagram (attributed to G. Higman) for each conjugacy class, depicting the action of PGL(2Z) on PL( F q).