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
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, 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 716777
• 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 05622792

External links

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