In mathematical group theory, the Suzuki group Suz or Sz is a sporadic simple group of order 2¹³ · 3⁷ · 5²· 7 · 11 · 13 = 448345497600 discovered by . It is not related to the Suzuki groups of Lie type.
The 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 6 and the 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.

Complex Leech lattice

The 24-dimensional 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...

 has a fixed-point-free automorphism of order 3. Identifying this with a complex cube root of 1 makes the Leech lattice into a 12 dimensional lattice over the Eisenstein integers, called the complex Leech lattice. The automorphism group of the complex Leech lattice is the universal cover 6.Suz of the Suzuki group Suz. This makes the group 6.Suz.2 into a maximal subgroup of Conway's group Co₀ = 2.Co₁of automorphisms of the Leech lattice, and shows that it has two complex irreducible representations of dimension 12. The group 6.Suz acting on the complex Leech lattice is analogous to the group 2.Co₁ acting on the Leech lattice.

Suzuki chain

The Suzuki chain or Suzuki tower is the following tower of rank 3 permutation group

Rank 3 permutation group

In mathematical finite group theory, a rank 3 permutation group is a group acting transitively on a set such that the stabilizer of a point has 3 orbits. The study of these groups was started by...

s from , each of which is the point stabilizer of the next.

• G₂(F₂)=U₃(3).2 has a rank 3 action on 36 = 1+14+21 points with point stabilizer PSL₃(F₂).2
• HJ.2 has a rank 3 action on 100 = 1+36+63 points with point stabilizer G₂(F₂)
• G₂(F₄).2 has a rank 3 action on 416 = 1+100+315 points with point stabilizer HJ.2
• Suz.2 has a rank 3 action on 1782=1+416+1365 points with point stabilizer G₂(F₄).2

Maximal subgroups

There are 17 conjugacy classes of maximal subgroups. Suz is the only sporadic group within Co₁ that inherits subgroups of order 13.

• G₂(4) order 251,596,800 index 1,782 (13 divides order)
• 3₂.U₄(3).2₃' order 19,595,520 index 22,880
• U₅(2) order 13,685,760 index 32,760
• 2¹⁺⁶.U₄(2) order 3,317,760 index 135,135
• 3⁵:M₁₁ order 1,924,560 index 232,960
• J₂:2 order 1,209,600 index 370,656
• 2⁴⁺⁶:3A₆ order 1,105,920 index 405,405
• (A₄ × L₃(4)):2 order 483,840 index 926,640
• 2^2+8:(A₅ × S₃) order 368,640 index 1,216,215
• M₁₂:2 order 190,080 index 2,358,720
• 3²⁺⁴:2.(A₄ × 2²).2 order 139,968 index 3,203,200
• (A₆ × A₅).2 order 43,200 index 10,378,368
• (A₆ × 3²:4).2 order 25,920 index 17,297,280
• L₃(3):2 order 11,232 index 39,916,800 (13 divides order)
• L₃(3):2
• L₂(25) order 7,800 index 57,480,192 (13 divides order)
• A₇ order 2,520 index 177,914,880