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.

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

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
2^1+6 · U(4, 2)	3,317,760	135,135
3^5 : M11	1,924,560	232,960
J2 : 2	1,209,600	370,656
2^4+6 : 3A6	1,105,920	405,405
(A4 × L3(4)) : 2	483,840	926,640
2^2+8 : (A5 × S3)	368,640	1,216,215
M12 : 2	190,080	2,358,720
3^2+4 : 2 · (A4 × 2^2) · 2	139,968	3,203,200
(A6 × A5) · 2	43,200	10,378,368
(A6 × 3^2 : 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

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

External links

• MathWorld: Suzuki group
• Atlas of Finite Group Representations: Suzuki group