Suzuki sporadic group

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

gollark: I don't know.

gollark: It would if stop was a negation, which it isn't, I think.

gollark: The processing is incredibly simplistic and won't do anything with complex clauses like that.

gollark: If that works properly it's purely by accident, I assure you.

gollark: ?charinfo Z

References

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.
Griess, Robert L. Jr. (1998), Twelve sporadic groups, Springer Monographs in Mathematics, Berlin, New York: Springer-Verlag, ISBN 978-3-540-62778-4, MR 1707296
Suzuki, Michio (1969), "A simple group of order 448,345,497,600", in Brauer, R.; Sah, Chih-han (eds.), Theory of Finite Groups (Symposium, Harvard Univ., Cambridge, Mass., 1968), Benjamin, New York, pp. 113–119, MR 0241527
Wilson, Robert A. (1983), "The complex Leech lattice and maximal subgroups of the Suzuki group", Journal of Algebra, 84 (1): 151–188, doi:10.1016/0021-8693(83)90074-1, ISSN 0021-8693, MR 0716777
Wilson, Robert A. (2009), The finite simple groups, Graduate Texts in Mathematics 251, 251, Berlin, New York: Springer-Verlag, doi:10.1007/978-1-84800-988-2, ISBN 978-1-84800-987-5, Zbl 1203.20012

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