Portal de Periódicos da CAPES

The transitive groups of degree 48 and some applications

2021; Elsevier BV; Volume: 607; Linguagem: Inglês

10.1016/j.jalgebra.2021.06.018

1090-266X

Derek F. Holt, Gordon Royle, Gareth Tracey,

Coding theory and cryptography

The primary purpose of this paper is to report on the successful enumeration in Magma of representatives of the 195826352 conjugacy classes of transitive subgroups of the symmetric group S48 of degree 48. In addition, we have determined that 25707 of these groups are minimal transitive and that 713 of them are elusive. The minimal transitive examples have been used to enumerate the vertex-transitive groups of degree 48, of which there are 1538868366, all but 0.1625% of which arise as Cayley graphs. We have also found that the largest number of elements required to generate any of these groups is 10, and we have used this fact to improve previous general bounds of the third author on the number of elements required to generate an arbitrary transitive permutation group of a given degree. The details of the proof of this improved bound will be published as a separate paper.