O'Nan group

In the area of abstract algebra known as group theory, the O'Nan group O'N or O'Nan–Sims group is a sporadic simple group of order
 * 29·34·5·73·11·19·31
 * = 460815505920
 * ≈ 5.

History
O'Nan is one of the 26 sporadic groups and was found by 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). The following simple groups have Sylow 2-subgroups of Alperin type:
 * For the Chevalley group G2(q), if q is congruent to 3 or 5 mod 8, n = 1 and the extension does not split.
 * For the Steinberg group 3D4(q), if q is congruent to 3 or 5 mod 8, n = 1 and the extension does not split.
 * For the alternating group A8, n = 1 and the extension splits.
 * For the O'Nan group, n = 2 and the extension does not split.
 * For the Higman-Sims group, n = 2 and the extension splits.

The Schur multiplier has order 3, and its outer automorphism group has order 2. showed that O'Nan cannot be a subquotient of the monster group. Thus it is one of the 6 sporadic groups called the pariahs.

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

Maximal subgroups
and independently found the 13 conjugacy classes of maximal subgroups of O'Nan as follows:


 * L3(7):2 (2 classes, fused by an outer automorphism)
 * J1 The subgroup fixed by an outer involution in O'Nan:2.
 * 42.L3(4):21 The centralizer of an (inner) involution in O'Nan.
 * (32:4 × A6).2
 * 34:21+4.D10
 * L2(31) (2 classes, fused by an outer automorphism)
 * 43.L3(2)
 * M11 (2 classes, fused by an outer automorphism)
 * A7 (2 classes, fused by an outer automorphism)

O'Nan moonshine
In 2017 John F. R. Duncan, Michael H. Mertens, and Ken Ono proved theorems that establish an analogue of monstrous moonshine for the O'Nan group. Their results "reveal a role for the O'Nan pariah group as a provider of hidden symmetry to quadratic forms and elliptic curves." The O'Nan moonshine results "also represent the intersection of moonshine theory with the Langlands program, which, since its inception in the 1960s, has become a driving force for research in number theory, geometry and mathematical physics." .

An informal description of these developments was written by in Quanta Magazine.