FACTOR GROUPS OF G(6, 6, 6) THROUGH COSET DIAGRAMS FOR AN ACTION ON PL(F-Q)

Let G=[x,y,t: x(2) = y(k) = t(2) = (xt)(2) = (yt)(2) = 1] and q be a prime power. Then any homomorphism from G into PGL(2,q) induces an action on the projective line over F,. Such an action can be depicted by a coset diagram. We show how the existence of certain types of fragments in these coset diagrams may be related to properties of a corresponding parameter I = r2/Delta, where r and Delta are the trace and determinant of a matrix representing the image of xy in PGL(2, q). We also show how these fragments can be used to show that for a family of positive integers n, all A(n) and S-n are quotients of G(6,6,6).