Suzuki sporadic group

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This article is about the sporadic simple group. For the infinite family of groups of Lie type found by Suzuki, see Suzuki groups.

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[edit]

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[edit]

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[edit]

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[edit]

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
21+6 · U(4, 2) 3,317,760 135,135
35 : M11 1,924,560 232,960
J2 : 2 1,209,600 370,656
24+6 : 3A6 1,105,920 405,405
(A4 × L3(4)) : 2 483,840 926,640
22+8 : (A5 × S3) 368,640 1,216,215
M12 : 2 190,080 2,358,720
32+4 : 2 · (A4 × 22) · 2 139,968 3,203,200
(A6 × A5) · 2 43,200 10,378,368
(A6 × 32 : 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[edit]

External links[edit]