An infinite Family of 4-Arc-Transitive Cubic Graphs Each with Girth 12

If p is any prime, and is that automorphism of the group SL(3,p) which takes each matrix to the transpose of its inverse,then there exists a connected trivalent graph (p) on vertices with the split extensionSL(3, p) as a group of automorphisms acting regularly on its4-arcs. In fact if p 3 then this group is the full automorphismgroup of (p), while the graph (3) is 5-arc-transitive with fullautomorphism group SL(3,3)0 x C₂. The girth of (p) is 12, exceptin th case p = 2 (where the girth is 6). Furthermore, in allcases (p) is bipartite, with SL(3, p) fixing each part. Alsowhen p 1 mod 3 the graph (p) is a triple cover of another trivalentgraph, which has automorphism group PSL(3, p)0 acting regularlyon its 4-arcs. These claims are proved using elementary theoryof symmetric graphs, together with a suitable choice of threematrices which generate SL(3, Z). They also provide a proofthat the group 4⁺(a¹²) described by Biggs in Computational grouptheor(ed. M. Atkinson) is infinite.