# Rudvalis group

In the mathematical field of group theory, the Rudvalis group Ru, found by Arunas Rudvalis (1973, 1984) and constructed by Conway and Wales (1973), is a sporadic simple group of order

214 · 33 · 53 ·· 13 · 29
= 145926144000
≈ 1011.

Ru is one of the six sporadic simple groups known as "pariah groups" as they are not found within the Monster group (Griess 1983, p. 91).

## Properties

The Rudvalis group acts as a rank 3 permutation group on 4060 points, with one point stabilizer the Ree group 2F4(2), the automorphism group of the Tits group. This representation implies a strongly regular graph in which each vertex has 2304 neighbors and 1755 non-neighbors. Two adjacent vertices have 1328 common neighbors; two non-adjacent ones have 1208 (Griess 1998, p. 125)

Its Schur multiplier has order 2, and its outer automorphism group is trivial.

Its double cover acts on a 28-dimensional lattice over the Gaussian integers. The lattice has 4×4060 minimal vectors; if minimal vectors are identified if one is 1, i, –1, or –i times another then the 4060 equivalence classes can be identified with the points of the rank 3 permutation representation. Reducing this lattice modulo the principal ideal

$(1 + i)\$

gives an action of the Rudvalis group on a 28-dimensional vector space over the field with 2 elements. Duncan (2006) used the 28-dimensional lattice to construct a vertex operator algebra acted on by the double cover.

Parrott (1976) characterized the Rudvalis group by the centralizer of a central involution. Aschbacher & Smith (2004) gave another characterization as part of their identification of the Rudvalis group as one of the quasithin groups.

## Maximal subgroups

Wilson (1984) found the 15 classes of maximal subgroups of the Rudvalis group, as follows: 2F4(2) = 2F4(2)'.2, 26.U3(3).2, (22 × Sz(8)):3, 23+8:L3(2), U3(5):2, 21+4+6.S5, PSL2(25).22, A8, PSL2(29), 52:4.S5, 3.A6.22, 51+2:[25], L2(13):2, A6.22, 5:4 × A5.