ATLAS: Linear group L_{2}(32)
Order = 32736 = 2^{5}.3.11.31.
Mult = 1.
Out = 5.
The following information is available for L_{2}(32):
Standard generators of L_{2}(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 L_{2}(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 > x^{16}.
An outer automorphism of L_{2}(32) can be obtained
by mapping (a, b) to
(a, b^{abababab}). This is in class 5A'' (ie, it is
conjugate to the field automorphism x > x^{4}).
Presentations for L_{2}(32) and L_{2}(32):5 in terms of their standard generators are given below.
< a, b  a^{2} = b^{3} = (ab)^{31} = (ab)^{4}(ab^{1}abab^{1})^{3}(ab)^{4}(ab^{1})^{2} = 1 >.
< c, d  c^{2} = d^{5} =
cdcdcd^{2}cd^{1}cd^{2}cdcd^{2}cd^{1}cd^{2}cd^{2} =
(cd)^{4}cd^{2}(cd^{2})^{3}cd^{2}cd^{2}cd^{1}cdcd^{2} = 1 >.
These presentations are available in Magma format as follows:
L2(32) on a and b and
L2(32):5 on c and d.
The representations of L_{2}(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 L_{2}(32):5 available are:

Permutations on 33 points:
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Permutations on 496 points:
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Permutations on 528 points:
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 10 over GF(2):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 20 over GF(2):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 20 over GF(2):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 32 over GF(2):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 40 over GF(2):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 40 over GF(2):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 80 over GF(2):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
The maximal subgroups of L_{2}(32) are as follows.
The maximal subgroups of L_{2}(32):5 are as follows.

L_{2}(32), with standard generators
c, (cddcdcdd)^1(cd)^5cddcdcdd.

2^{5}:31:5 = F_{4960}, with generators
(cd)^1dcd, (cdd)^2dcddcdd,
and with generators cdcd^3cdc, d.

33:10 = F_{330}, with generators
c, (cdcdd)^4cd(cdcdd)^4,
and with generators c, ddcddcdcd^1.

31:10 = F_{310}, with generators
(cd)^2dcd, (cdd)^2d(cdd)^2,
and with generators c, ddcdcdcd^1.
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'''].
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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.