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

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. 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 = U · 2 has a rank 3 action on 36 = 1 + 14 + 21 points with point stabilizer PSL · 2
  • J2 · 2 has a rank 3 action on 100 = 1 + 36 + 63 points with point stabilizer G2
  • G2 · 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 · 2

    Maximal subgroups

found the 17 conjugacy classes of maximal subgroups of Suz as follows:
No.StructureOrderIndexComments
1G2251,596,800
= 212·33·52·7·13		1,782
= 2·34·11
232· U : 2'319,595,520
= 28·37·5·7		22,880
= 25·5·11·13		normalizer of a subgroup of order 3
3U13,685,760
= 210·35·5·11		32,760
= 23·32·5·7·13
42 · U3,317,760
= 213·34·5		135,135
= 33·5·7·11·13		centralizer of an involution of class 2A
535 : M111,924,560
= 24·37·5·11		232,960
= 29·5·7·13
6J2 : 21,209,600
= 28·33·52·7		370,656
= 25·3^4·11·13		the subgroup fixed by an outer involution of class 2C
724+6 : 3A61,105,920
= 213·33·5		405,405
= 34·5·7·11·13
8 : 2483,840
= 29·33·5·7		926,640
= 24·34·5·11·13
922+8 : 368,640
= 213·32·5		1,216,215
= 35·5·7·11·13
10M12 : 2190,080
= 27·33·5·11		2,358,720
= 26·34·5·7·13		the subgroup fixed by an outer involution of class 2D
1132+4 : 2.2139,968
= 26·37		3,203,200
= 27·52·7·11·13
12 · 243,200
= 26·33·52		10,378,368
= 27·3^4·7·11·13
13 · 225,920
= 26·34·5		17,297,280
= 27·33·5·7·11·13
14,15L3 : 211,232
= 25·33·13		39,916,800
= 28·34·5^2·7·11		two classes, fused by an outer automorphism
16L27,800
= 23·3·52·13		57,480,192
= 210·36·7·11
17A72,520
= 23·32·5·7		177,914,880
= 210·35·5·11·13