|Algebraic structure → Group theory
- 29 · 34 · 5 · 73 · 11 · 19 · 31
- = 460815505920
The Schur multiplier has order 3, and its outer automorphism group has order 2.
The O'Nan group was found by Michael O'Nan (1976) in a study of groups with a Sylow 2-subgroup of "Alperin type", meaning isomorphic to a Sylow 2-Subgroup of a group of type (Z/2nZ ×Z/2nZ ×Z/2nZ).PSL3(F2). For the O'Nan group n=2 and the extension does not split. The only other simple group with a Sylow 2-subgroup of Alperin type with n≥2 is the Higman–Sims group again with n=2, but the extension splits.
Ryba (1988) showed that its triple cover has two 45-dimensional representations over the field with 7 elements, exchanged by an outer automorphism.
L3(7):2 (2 classes, fused by an outer automorphism)
J1 The subgroup fixed by an outer involution.
(32:4 × A6).2
L2(31) (2 classes, fused by an outer automorphism)
M11 (2 classes, fused by an outer automorphism)
A7 (2 classes, fused by an outer automorphism)
- O'Nan, Michael E. (1976), Some evidence for the existence of a new simple group, Proceedings of the London Mathematical Society. Third Series 32 (3): 421–479, doi:10.1112/plms/s3-32.3.421, ISSN 0024-6115, MR 0401905
- R. L. Griess, Jr, The Friendly Giant, Inventiones Mathematicae 69 (1982), 1-102. p. 94: proof that O'N is a pariah.
- A. J. E. Ryba, A new construction of the O'Nan simple group. J. Algebra 112 (1988), no. 1, 173-197.MR 0921973
- Wilson, Robert A. (1985), The maximal subgroups of the O'Nan group, Journal of Algebra 97 (2): 467–473, doi:10.1016/0021-8693(85)90059-6, ISSN 0021-8693, MR 812997
- Yoshiara, Satoshi (1985), The maximal subgroups of the sporadic simple group of O'Nan, Journal of the Faculty of Science. University of Tokyo. Section IA. Mathematics 32 (1): 105–141, ISSN 0040-8980, MR 783183
- Atlas of Finite Group Representations: O'Nan group