In the mathematical discipline known as group theory

Group theory

In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and...

, the McLaughlin group McL is a sporadic simple group

Simple group

In mathematics, a simple group is a nontrivial group whose only normal subgroups are the trivial group and the group itself. A group that is not simple can be broken into two smaller groups, a normal subgroup and the quotient group, and the process can be repeated...

 of order 2⁷ · 3⁶ · 5³· 7 · 11 = 898,128,000, discovered by as an index 2 subgroup of a rank 3 permutation group acting on the McLaughlin graph with 275 =1+112+162 vertices. It fixes a 2-2-3 triangle in the Leech lattice

Leech lattice

In mathematics, the Leech lattice is an even unimodular lattice Λ24 in 24-dimensional Euclidean space E24 found by .-History:Many of the cross-sections of the Leech lattice, including the Coxeter–Todd lattice and Barnes–Wall lattice, in 12 and 16 dimensions, were found much earlier than...

 so is a subgroup of the Conway group

Conway group

In mathematics, the Conway groups Co1, Co2, and Co3 are three sporadic groups discovered by John Horton Conway.The largest of the Conway groups, Co1, of order...

s.
Its Schur multiplier

Schur multiplier

In mathematical group theory, the Schur multiplier or Schur multiplicator is the second homology group H_2 of a group G.It was introduced by in his work on projective representations.-Examples and properties:...

 has order 3, and its outer automorphism group

Outer automorphism group

In mathematics, the outer automorphism group of a group Gis the quotient Aut / Inn, where Aut is the automorphism group of G and Inn is the subgroup consisting of inner automorphisms. The outer automorphism group is usually denoted Out...

 has order 2. The group 3.McL.2 is a maximal subgroup of the Lyons group

Lyons group

In the mathematical field of group theory, the Lyons group Ly , is a sporadic simple group of order...

.

McL has one conjugacy class of involution (element of order 2), whose centralizer is an interesting maximal subgroup of type 2.A₈. This has a center of order 2; the quotient modulo the center is isomorphic to the alternating group A₈.

In the Conway group

Conway group

In mathematics, the Conway groups Co1, Co2, and Co3 are three sporadic groups discovered by John Horton Conway.The largest of the Conway groups, Co1, of order...

Co₃, McL has the normalizer McL:2, which is maximal in Co₃.

McL is the only sporadic group to admit irreducible representations of quaternionic type. It has 2 such representations, one of dimension 3520 and one of dimension 4752.

Maximal subgroups

There are 12 conjugacy classes of maximal subgroups.

• U₄(3) order 3,265,920 index 275
• M₂₂ order 443,520 index 2,025
• M₂₂
• U₃(5) order 126,000 index 7,128
• 3¹⁺⁴:2.S₅ order 58,320 index 15,400
• 3⁴:M₁₀ order 58,320 index 15,400
• L₃(4):2₂ order 40,320 index 22,275
• 2.A₈ order 40,320 index 22,275 - centralizer of involution
• 2⁴:A₇ order 40,320 index 22,275
• 2⁴:A₇
• M₁₁ order 7,920 index 113,400
• 5₊¹⁺²:3:8 order 3,000 index 299,376

The 2 pairs of isomorphic classes both fuse in the larger group McL:2.