In the area of modern algebra known as group theory, the Suzuki group Suz or Sz is a sporadic simple group of order

213 · 37 · 52 · 7 · 11 · 13 = 448345497600
≈ 4×1011.

## History

Suz is one of the 26 Sporadic groups and was discovered by Suzuki (1969) as a rank 3 permutation group on 1782 points with point stabilizer G2(4). It is not related to the Suzuki groups of Lie type. The Schur multiplier has order 6 and the outer automorphism group has order 2.

## Complex Leech lattice

The 24-dimensional Leech lattice 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. This makes the group 6 · Suz · 2 into a maximal subgroup of Conway's group Co0 = 2 · Co1 of 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 groups from (Suzuki 1969), each of which is the point stabilizer of the next.

• G2(2) = U(3, 3) · 2 has a rank 3 action on 36 = 1 + 14 + 21 points with point stabilizer PSL(3, 2) · 2
• J2 · 2 has a rank 3 action on 100 = 1 + 36 + 63 points with point stabilizer G2(2)
• G2(4) · 2 has a rank 3 action on 416 = 1 + 100 + 315 points with point stabilizer J2 · 2
• Suz · 2 has a rank 3 action on 1782 = 1 + 416 + 1365 points with point stabilizer G2(4) · 2

## Maximal subgroups

Wilson (1983) found the 17 conjugacy classes of maximal subgroups of Suz as follows:

Maximal Subgroup Order Index
G2(4) 251,596,800 1782
32 · U(4, 3) · 23 19,595,520 22,880
U(5, 2) 13,685,760 32,760
21+6 · U(4, 2) 3,317,760 135,135
35 : M11 1,924,560 232,960
J2 : 2 1,209,600 370,656
24+6 : 3A6 1,105,920 405,405
(A4 × L3(4)) : 2 483,840 926,640
22+8 : (A5 × S3) 368,640 1,216,215
M12 : 2 190,080 2,358,720
32+4 : 2 · (A4 × 22) · 2 139,968 3,203,200
(A6 × A5) · 2 43,200 10,378,368
(A6 × 32 : 4) · 2 25,920 17,297,280
L3(3) : 2 11,232 39,916,800
L2(25) 7,800 57,480,192
A7 2,520 177,914,880