# ATLAS: Suzuki group Suz

Order = 448345497600 = 213.37.52.7.11.13.
Mult = 6.
Out = 2.

The following information is available for Suz:

### Standard generators

Standard generators of the Suzuki group Suz are a and b where a is in class 2B, b is in class 3B, ab has order 13 and ababb has order 15.
Standard generators of 2.Suz are preimages A and B where B has order 3 and AB has order 13.
Standard generators of 3.Suz are preimages A and B where A has order 2 and AB has order 13.
Standard generators of 6.Suz are preimages A and B where A has order 4, B has order 3 and AB has order 13.

Standard generators of the automorphism group Suz:2 are c and d where c is in class 2C, d is in class 3B and cd has order 28.
Standard generators of 2.Suz:2 are preimages C and D where D has order 3.
Standard generators of 3.Suz:2 are preimages C and D where D is in class +3B (equivalently, CDCDD has order 7).
Standard generators of 6.Suz:2 are preimages C and D where D has order 3 and CDCDD has order 7 or 14. (The order of CDCDD is determined by the sixfold cover.)

### Automorphisms

The outer automorphism of Suz may be realised by mapping a, b to aabab, babbabb.
If c' is the 15th power of this automorphism, and d' = b, then (c', d') is conjugate to (c, d).
Another outer automorphism [with shorter words] maps a, b to a, bababbab.
If u is this second automorphism, then u is class 8H and ((babu)7, baba) is conjugate to (c, d).

### Black box algorithms

#### Finding generators

To find standard generators for Suz:

• Find any element of order 14. Its 7th power is a 2B-element, x say.
[The probability of success at each attempt is 1 in 28.]
• Find any element of order 9 or 18. This powers up to a 3B-element, y say.
[The probability of success at each attempt is 4 in 27 (about 1 in 7).]
• Find a conjugate a of x and a conjugate b of y such that ab has order 13 and ababb has order 15.
[The probability of success at each attempt is 18 in 715 (about 1 in 40).]
This algorithm is available in computer readable format: finder for Suz.

To find standard generators for Suz.2:

• Find any element of order 30. It powers up to a 2C-element, x say.
[The probability of success at each attempt is 1 in 30.]
• Find any element of order 9 or 18. This powers up to a 3B-element, y say.
[The probability of success at each attempt is 2 in 27 (about 1 in 14).]
• Find a conjugate c of x and a conjugate d of y whose product has order 28.
[The probability of success at each attempt is 27 in 143 (about 1 in 5).]
This algorithm is available in computer readable format: finder for Suz.2.

#### Checking generators

To check that elements x and y of Suz are standard generators:

• Check o(x) = 2
• Check o(y) = 3
• Check o(xy) = 13
• Check o(xyxyy) = 15
• Check o(xyxyxyy) = 12
This algorithm is available in computer readable format: checker for Suz.

To check that elements x and y of Suz.2 are standard generators:

• Check o(x) = 2
• Check o(y) = 3
• Check o(xy) = 28
• Check o(xyxyxyyxyy) = 7
This algorithm is available in computer readable format: checker for Suz.2

### Representations

Representations are available for the following decorations of Suz: The representations of Suz available are:
The representations of 2.Suz available are:
The representations of 3.Suz available are:
The representations of 6.Suz available are:
The representations of Suz:2 available are:
The representations of 2.Suz:2 available are:
The representations of 3.Suz:2 available are:
The representations of 6.Suz:2 available are:
• Dimension 24 over GF(3): C and D (Meataxe), C and D (Meataxe binary), C and D (GAP). - a reducible representation
• Dimension 24 over GF(5): C and D (Meataxe), C and D (Meataxe binary), C and D (GAP).
• Dimension 24 over GF(11): C and D (Meataxe), C and D (Meataxe binary), C and D (GAP).

### Maximal subgroups

The maximal subgroups of Suz are:
The maximal subgroups of Suz:2 are:

### Conjugacy classes

A set of generators for the maximal cyclic subgroups can be obtained by running this program on the standard generators. All conjugacy classes can therefore be obtained as suitable powers of these elements. Problems of algebraic conjugacy are not yet dealt with. Go to main ATLAS (version 2.0) page. Go to sporadic groups page. Go to old Suz page - ATLAS version 1. Anonymous ftp access is also available. See here for details.

Version 2.0 created on 20th December 2000.
Last updated 7.1.05 by SJN.
Information checked to Level 0 on 20.12.00 by RAW.
R.A.Wilson, R.A.Parker and J.N.Bray.