A note on Cayley graphs (Q1057873)

An example is given of a finite group A of order 144, with a generating set \(X=\{x,y\}\) such that \(x^ 3=y^ 2=1\) and such that the Cayley graph C(A,X) has genus 4 and characteristic -6 (both of which are small relative to the order of A), although there is no short relator of the form \((xy)^ r\) with \(r<12\) or of the form \([x,y]^ r\) with \(r<6\). Accordingly this and other possible examples do not fit into a pattern suggested by Tom Tucker's refined Hurwitz theorem for imbeddings of Cayley graphs.