A list is given of all semisymmetric (edge-but not vertex-transitive) connected finite cubic graphs of order up to 768. This list was determined by the authors using Goldschmidt's classification of finite primitive amalgams of index (3, 3), and a computer algorithm for finding all normal subgroups of up to a given index in a finitely-presented group. The list includes several previously undiscovered graphs. For each graph in the list, a significant amount of information is provided, including its girth and diameter, the order of its automorphism group, the order and structure of a minimal edge-transitive group of automorphisms, its Goldschmidt type, stabiliser partitions, and other details about its quotients and covers. A summary of all known infinite families of semisymmetric cubic graphs is also given, together with explicit rules for their construction, and members of the list are identified with these. The special case of those graphs having K 1,3 as a normal quotient is investigated in detail.

This paper describes the determination of all orientably-regular maps and hypermaps of genus 2 to 101, and all non-orientable regular maps and hypermaps of genus 3 to 202. It extends the lists obtained by Conder and Dobcsányi (2001) of all such maps of Euler characteristic −1 to −28, and corrects errors made in those lists for the vertex-or face-multiplicities of 14 'cantankerous' non-orientable regular maps. Also some discoveries are announced about the genus spectrum of orientably-regular but chiral maps, and the genus spectrum of orientably-regular maps having no multiple edges, made possible by observations of patterns in the extension of these lists to higher genera.

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ABSTRACT. Hurwitz groups are the nontrivial finite quotients 2 3 7of the (2,3,7) triangle group (x, y\x -y = (xy) = 1) . This paper gives a brief survey of such groups, their significance, and some of their properties, together with a description of all examples known to the author.

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A graph Γ is symmetric if its automorphism group acts transitively on the arcs of Γ , and s-regular if its automorphism group acts regularly on the set of s-arcs of Γ . Tutte (1947, 1959) showed that every finite symmetric cubic graph is s-regular for some s 5. Djoković and Miller (1980) proved that there are seven types of arc-transitive group action on finite cubic graphs, characterised by the stabilisers of a vertex and an edge. A given finite symmetric cubic graph, however, may admit more than one type of arctransitive group action. In this paper we determine exactly which combinations of types are possible. Some combinations are easily eliminated by existing theory, and others can be eliminated by elementary extensions of that theory. The remaining combinations give 17 classes of finite symmetric cubic graph, and for each of these, we prove the class is infinite, and determine at least one representative. For at least 14 of these 17 classes the representative we give has the minimum possible number of vertices (and we show that in two of these 14 cases every graph in the class is a cover of the smallest representative), while for the other three classes, we give the smallest examples known to us. In an appendix, we give a table showing the class of every symmetric cubic graph on up to 768 vertices.