# ATLAS: Linear group L2(32)

Order = 32736 = 25.3.11.31.
Mult = 1.
Out = 5.

The following information is available for L2(32):

### Standard generators

Standard generators of L2(32) are a and b where a has order 2, b has order 3 and ab is in class 31A/B/C/D/E. The last condition is equivalent to saying that ab has order 31 and ababb has order 11.
Standard generators of L2(32):5 are c and d where c has order 2, d has order 5, cd, cdd, cdcdd, cdcdcdd each have order 15, and cdcdcdcddcdcddcdd has order 11.
The conditions ensure that d is conjugate to the field automorphism x -> x16.

### Automorphisms

An outer automorphism of L2(32) can be obtained by mapping (a, b) to (a, babababab). This is in class 5A'' (ie, it is conjugate to the field automorphism x -> x4).

### Presentations

Presentations for L2(32) and L2(32):5 in terms of their standard generators are given below.

< a, b | a2 = b3 = (ab)31 = (ab)4(ab-1abab-1)3(ab)4(ab-1)2 = 1 >.

< c, d | c2 = d5 = cdcdcd-2cd-1cd-2cdcd-2cd-1cd2cd-2 = (cd)4cd-2(cd2)3cd-2cd2cd-1cdcd2 = 1 >.

These presentations are available in Magma format as follows: L2(32) on a and b and L2(32):5 on c and d.

### Representations

The representations of L2(32) available are:
• Permutations on 33 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Permutations on 496 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Permutations on 528 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 2a over GF(32): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP). - the natural representation.
• Dimension 31a over GF(3): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 31b over GF(11): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 32 over GF(31): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 32 over Z: a and b (Magma).
• Dimension 33a over Z[z31]: a and b (Magma). - monomial.
The representations of L2(32):5 available are:

### Maximal subgroups

The maximal subgroups of L2(32) are as follows.
The maximal subgroups of L2(32):5 are as follows.

### Conjugacy classes

A set of generators for the maximal cyclic subgroups of L2(32) can be obtained by running this program on the standard generators. All conjugacy classes can therefore be obtained as suitable powers of these elements.

A set of generators for the maximal cyclic subgroups of L2(32):5 can be obtained by running this program on the standard generators. All conjugacy classes can therefore be obtained as suitable powers of these elements. The given class representatives for classes 10A and 15A lie in the same coset of L2(32) as d. We thus take d to lie in class 5A [not 5A', 5A'' or 5A''']. Go to main ATLAS (version 2.0) page. Go to linear groups page. Go to old L2(32) page - ATLAS version 1. Anonymous ftp access is also available. See here for details.

Version 2.0 file created on 17th January 2002, from Version 1 file last modified on 07.10.96.
Last updated 01.02.02 by JNB.
Information checked to Level 0 on 17.01.02 by RAW.
R.A.Wilson, R.A.Parker and J.N.Bray.