# ATLAS: Linear group L2(31)

Order = 14880 = 25.3.5.31.
Mult = 2.
Out = 2.

### Standard generators

Standard generators of L2(31) are a and b where a has order 2, b has order 3 and ab has order 31.
Standard generators of the double cover 2.L2(31) = SL2(31) are preimages A and B where B has order 3 and AB has order 31.

Standard generators of L2(31):2 = PGL2(31) are c and d where c has order 2, d has order 3, cd has order 10 and cdcdd has order 8.
Standard generators of either of the double covers 2.L2(31).2 = 2.PGL2(31) are preimages C and D where D has order 3.

### Presentations

Presentations for L2(31) and L2(31):2 = PGL2(31) in terms of their standard generators are given below.

< a, b | a2 = b3 = [a,b]3ab[a,b][a,bab][a,b-1abab] = 1 >.

< c, d | c2 = d3 = (cd)10 = [c, d]8 = (cd(cdcdcd-1)3)2 = 1 >.

### Representations

The representations of L2(31) available are:
• Permutations on 32 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Some matrix representations in characteristic 2:
• Dimension 31 over GF(3): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 31 over GF(5): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 3 over GF(31): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 31 over GF(31): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Some matrix representations in characteristic 0:
The representations of 2.L2(31) = SL2(31) available are
• Dimension 2 over GF(31): A and B (Meataxe), A and B (Meataxe binary), A and B (GAP).
• Dimension 16 over GF(5): A and B (Meataxe), A and B (Meataxe binary), A and B (GAP).
• Some matrix representations in characteristic 0:
• Dimension 32 over Z(z30) (monomial): A and B (GAP).
The representations of L2(31):2 available are:

### Maximal subgroups

The maximal subgroups of L2(31) are as follows.
• 31:15, with generators ???.
• A5
• A5
• D32
• D30
• S4
• S4
The maximal subgroups of L2(31):2 are as follows.
• L2(31)
• 31:30
• D64
• D60

### Conjugacy classes

A set of generators for the maximal cyclic subgroups of L2(31) 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(31):2 can be obtained by running this program on the standard generators. All conjugacy classes can therefore be obtained as suitable powers of these elements. Go to main ATLAS (version 2.0) page. Go to linear groups page. Go to old L2(31) 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 28.10.98.
Last updated 27.06.06 by JNB.
Information checked to Level 0 on 17.01.02 by RAW.
R.A.Wilson, R.A.Parker and J.N.Bray.