Factor groups of G 6,6,6 through coset diagrams for an action on PL ( F q)

Abstract

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 q . 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 ϑ= r 2/ βdG, where r and βdG 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 .