An Infinite Family of 4-arc-transitive Cubic Graphs Each with Girth 12

If p is any prime, and 6 is that automorphism of the group SL(3,/>) which takes each matrix to the transpose of its inverse, then there exists a connected trivalent graph F(p) on ^p(p—\)(p — \) vertices with the split extension SL(3,/?)<0> as a group of automorphisms acting regularly on its 4-arcs. In fact if/? / 3 then this group is the full automorphism group of f(p), while the graph F(3) is 5-arc-transitive with full automorphism group SL(3,3)<0> x C2. The girth of F(p) is 12, except in the case p = 2 (where the girth is 6). Furthermore, in all cases F(p) is bipartite, with SL(3,p) fixing each part. Also when p = 1 mod 3 the graph T(p) is a triple cover of another trivalent graph, which has automorphism group PSL(3,p)<0> acting regularly on its 4-arcs. These claims are proved using elementary theory of symmetric graphs, together with a suitable choice of three matrices which generate SL(3, Z). They also provide a proof that the group 4(a) described by Biggs in Computational group theory (ed. M. Atkinson) is infinite.