# Suzuki sporadic group

Algebraic structure → Group theory Group theory :
Basic notions
Subgroup
Normal subgroup
Group action
Quotient group
(Semi-)direct product
Direct sum
Free product
Wreath product
Group homomorphisms
kernel
image
simple
finite
infinite
continuous
multiplicative
additive
cyclic
abelian
dihedral
nilpotent
solvable
Glossary of group theory
List of group theory topics :
Finite groups
Cyclic group Z$_n$
Symmetric group S$_n$
Alternating group A$_n$
Dihedral group D$_n$
Quaternion group Q
Cauchy's theorem
Lagrange's theorem
Sylow theorems
Hall's theorem
p-group
Elementary abelian group
Frobenius group
Schur multiplier
Classification of finite simple groups
cyclic
alternating
Lie type
sporadic :
Discrete groupsLattices
Integers ($\mathbb {Z}$)
Free group
Modular groups PSL(2, $\mathbb {Z}$)SL(2, $\mathbb {Z}$)
Arithmetic group
Lattice
Hyperbolic group :
Topological and Lie groups
Solenoid
Circle
General linear GL(n)
Special linear SL(n)
Orthogonal O(n)
Euclidean E(n)
Special orthogonal SO(n)
Unitary U(n)
Special unitary SU(n)
Symplectic Sp(n)
G$_2$
F$_4$
E$_6$
E$_7$
E$_8$
Lorentz
Poincaré
Conformal
Diffeomorphism
Loop
Infinite dimensional Lie group O(∞)SU(∞)Sp(∞) :
Algebraic groups
Linear algebraic group
Reductive group
Abelian variety
Elliptic curve :
vte :
In the area of modern algebra known as group theory, the Suzuki group Suz or Sz is a sporadic simple group of order
    448,345,497,600 = 2$^{13}$ · 3$^7$ · 5$^2$ · 7 · 11 · 13 ≈ 4×10$^{11}$.

## 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 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.

## 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 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.

## 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.
* G$_2$(2) = U(3, 3) · 2 has a rank 3 action on 36 = 1 + 14 + 21 points with point stabilizer PSL(3, 2) · 2
* J$_2$ · 2 has a rank 3 action on 100 = 1 + 36 + 63 points with point stabilizer G$_2$(2)
* G$_2$(4) · 2 has a rank 3 action on 416 = 1 + 100 + 315 points with point stabilizer J$_2$ · 2
* Suz · 2 has a rank 3 action on 1782 = 1 + 416 + 1365 points with point stabilizer G$_2$(4) · 2

## Maximal subgroups

Wilson (1983) found the 17 conjugacy classes of maximal subgroups of Suz as follows:
Maximal subgroups of Suz :
| No. | Structure | Order | Index | Comments |
| 1 | G$_2$(4) | 251,596,800 = 2$^{12}$·3$^3$·5$^2$·7·13 | 1,782 = 2·3$^4$·11 | |
| 2 | 3$_2$$^·$U(4, 3) : 2'$_3$ | 19,595,520 = 2$^8$·3$^7$·5·7 | 22,880 = 2$^5$·5·11·13 | normalizer of a subgroup of order 3 (class 3A) |
| 3 | U(5, 2) | 13,685,760 = 2$^{10}$·3$^5$·5·11 | 32,760 = 2$^3$·3$^2$·5·7·13 | |
| 4 | 2$^{1+6}$ $_–$$^·$U(4, 2) | 3,317,760 = 2$^{13}$·3$^4$·5 | 135,135 = 3$^3$·5·7·11·13 | centralizer of an involution of class 2A |
| 5 | 3$^5$ : M$_{11}$ | 1,924,560 = 2$^4$·3$^7$·5·11 | 232,960 = 2$^9$·5·7·13 | |
| 6 | J$_2$ : 2 | 1,209,600 = 2$^8$·3$^3$·5$^2$·7 | 370,656 = 2$^5$·3^4·11·13 | the subgroup fixed by an outer involution of class 2C |
| 7 | 2$^{4+6}$ : 3A$_6$ | 1,105,920 = 2$^{13}$·3$^3$·5 | 405,405 = 3$^4$·5·7·11·13 | |
| 8 | (A$_4$ × L$_3$(4)) : 2 | 483,840 = 2$^9$·3$^3$·5·7 | 926,640 = 2$^4$·3$^4$·5·11·13 | |
| 9 | 2$^{2+8}$ : (A$_5$ × S$_3$) | 368,640 = 2$^{13}$·3$^2$·5 | 1,216,215 = 3$^5$·5·7·11·13 | |
| 10 | M$_{12}$ : 2 | 190,080 = 2$^7$·3$^3$·5·11 | 2,358,720 = 2$^6$·3$^4$·5·7·13 | the subgroup fixed by an outer involution of class 2D |
| 11 | 3$^{2+4}$ : 2(A$_4$ × 2$^2$).2 | 139,968 = 2$^6$·3$^7$ | 3,203,200 = 2$^7$·5$^2$·7·11·13 | |
| 12 | (A$_6$ × A$_5$) · 2 | 43,200 = 2$^6$·3$^3$·5$^2$ | 10,378,368 = 2$^7$·3^4·7·11·13 | |
| 13 | (A$_6$ × 3$^2$ : 4)$^·$2 | 25,920 = 2$^6$·3$^4$·5 | 17,297,280 = 2$^7$·3$^3$·5·7·11·13 | |
| 14,15 | L$_3$(3) : 2 | 11,232 = 2$^5$·3$^3$·13 | 39,916,800 = 2$^8$·3$^4$·5^2·7·11 | two classes, fused by an outer automorphism |
| 16 | L$_2$(25) | 7,800 = 2$^3$·3·5$^2$·13 | 57,480,192 = 2$^{10}$·3$^6$·7·11 | |
| 17 | A$_7$ | 2,520 = 2$^3$·3$^2$·5·7 | 177,914,880 = 2$^{10}$·3$^5$·5·11·13 | |