A census of small transitive groups and vertex-transitive graphs. (English) Zbl 1528.20004
Summary: We describe two similar but independently-coded computations used to construct a complete catalogue of the transitive groups of degree less than 48, thereby verifying, unifying and extending the catalogues previously available. From this list, we construct all the vertex-transitive graphs of order less than 48. We then present a variety of summary data regarding the transitive groups and vertex-transitive graphs, focusing on properties that seem to occur most frequently in the study of groups acting on graphs. We illustrate how such catalogues can be used, first by finding a complete list of the elusive groups of order at most 47 and then by completely determining which groups of order at most 47 are CI groups.
MSC:
20B20 | Multiply transitive finite groups |
05C25 | Graphs and abstract algebra (groups, rings, fields, etc.) |
05E18 | Group actions on combinatorial structures |
20-08 | Computational methods for problems pertaining to group theory |
Keywords:
transitive permutation group; vertex-transitive graph; computer enumeration; Cayley graph; elusive group; CI-groupOnline Encyclopedia of Integer Sequences:
Number of transitive permutation groups of degree n.Number of vertex-transitive graphs with n nodes.
Number of Cayley graphs on n nodes.
Number of minimal transitive permutation groups of degree n.
References:
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