# ATLAS: Linear group L2(32)

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

### 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 (for the moment at least) 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.

### 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 = (cd)?? = ... = 1 >.

### Representations

The representations of L2(32) available are
• a and b as permutations on 33 points.
• a and b as 2 x 2 matrices over GF(32) - the natural representation.
The representations of L2(32).5 available are
• c and d as 10 x 10 matrices over GF(2).

### Maximal subgroups

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