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Suzuki sporadic group

Sporadic simple group

Note

the sporadic simple group

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.

History

Suz is one of the 26 Sporadic groups and was discovered by 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 , 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

found the 17 conjugacy classes of maximal subgroups of Suz as follows:

No.	Structure	Order
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	2 · 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

Bibliography

Conway, J. H.; Curtis, R. T.; Norton, S. P.; Parker, R. A.; and Wilson, R. A.: "Atlas of Finite Groups: Maximal Subgroups and Ordinary Characters for Simple Groups." Oxford, England 1985.

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