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In the area of modern algebra known as group theory, the Suzuki group Suz or Sz is a sporadic simple group of order
Contents
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.
Maximal subgroups
Wilson (1983) found the 17 conjugacy classes of maximal subgroups of Suz as follows: