Suzuki sporadic group

This article is about the sporadic simple group. For the infinite family of groups of Lie type found by Suzuki, see Suzuki groups.

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

   448,345,497,600 = 2¹³ · 3⁷ · 5² · 7 · 11 · 13 ≈ 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 subgroups of Suz

No.	Structure	Order	Index	Comments
1	G2(4)	251,596,800 = 212·33·52·7·13	1,782 = 2·34·11
2	32· U(4, 3) : 2'3	19,595,520 = 28·37·5·7	22,880 = 25·5·11·13	normalizer of a subgroup of order 3 (class 3A)
3	U(5, 2)	13,685,760 = 210·35·5·11	32,760 = 23·32·5·7·13
4	21+6  – · U(4, 2)	3,317,760 = 213·34·5	135,135 = 33·5·7·11·13	centralizer of an involution of class 2A
5	35 : M11	1,924,560 = 24·37·5·11	232,960 = 29·5·7·13
6	J2 : 2	1,209,600 = 28·33·52·7	370,656 = 25·3^4·11·13	the subgroup fixed by an outer involution of class 2C
7	24+6 : 3A6	1,105,920 = 213·33·5	405,405 = 34·5·7·11·13
8	(A4 × L3(4)) : 2	483,840 = 29·33·5·7	926,640 = 24·34·5·11·13
9	22+8 : (A5 × S3)	368,640 = 213·32·5	1,216,215 = 35·5·7·11·13
10	M12 : 2	190,080 = 27·33·5·11	2,358,720 = 26·34·5·7·13	the subgroup fixed by an outer involution of class 2D
11	32+4 : 2(A4 × 22).2	139,968 = 26·37	3,203,200 = 27·52·7·11·13
12	(A6 × A5) · 2	43,200 = 26·33·52	10,378,368 = 27·3^4·7·11·13
13	(A6 × 32 : 4) · 2	25,920 = 26·34·5	17,297,280 = 27·33·5·7·11·13
14,15	L3(3) : 2	11,232 = 25·33·13	39,916,800 = 28·34·5^2·7·11	two classes, fused by an outer automorphism
16	L2(25)	7,800 = 23·3·52·13	57,480,192 = 210·36·7·11
17	A7	2,520 = 23·32·5·7	177,914,880 = 210·35·5·11·13