# 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

*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 G_{2}(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.

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 Co_{0} = 2 · Co_{1} 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 · Co_{1} acting on the Leech lattice.

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.

Wilson (1983) found the 17 conjugacy classes of maximal subgroups of *Suz* as follows: