O'Nan group

In the area of modern algebra known as group theory, the O'Nan group O'N or O'Nan–Sims group is a sporadic simple group of order

History

O'N is one of the 26 sporadic groups and 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.

The Schur multiplier has order 3, and its outer automorphism group has order 2.

O'N is one of the 6 sporadic simple groups called the pariahs because (Griess 1982) showed that it is not a subquotient of the monster group.

Representations

Ryba (1988) showed that its triple cover has two 45-dimensional representations over the field with 7 elements, exchanged by an outer automorphism.

Maximal subgroups

Wilson (1985) and Yoshiara (1985) independently found the 13 conjugacy classes of maximal subgroups of O'N as follows: