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.
found the 17 conjugacy classes of maximal subgroups of Suz as follows:
Maximal SubgroupOrderIndex
G2251,596,8001782
32 · U · 2319,595,52022,880
U13,685,76032,760
21+6 · U3,317,760135,135
35 : M111,924,560232,960
J2 : 21,209,600370,656
24+6 : 3A61,105,920405,405
: 2483,840926,640
22+8 : 368,6401,216,215
M12 : 2190,0802,358,720
32+4 : 2 · · 2139,9683,203,200
· 243,20010,378,368
· 225,92017,297,280
L3 : 211,23239,916,800
L27,80057,480,192
A72,520177,914,880