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: 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'''].


Main ATLAS page Go to main ATLAS (version 2.0) page.
Linear groups page Go to linear groups page.
Old L2(32) page Go to old L2(32) page - ATLAS version 1.
ftp access 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.