# ATLAS: Linear group L3(7)

Order = 1876896.
Mult = 3.
Out = S3.

The following information is available for L3(7):

### Standard generators

Standard generators of L3(7) are a and b where a has order 2, b has order 3, ab has order 19, and ababb has order 6.
Standard generators of 3.L3(7) are preimages A and B where A has order 2 and AB has order 19.

Standard generators of L3(7).2 are c and d where c is in class 2B, d is in class 4B, cd has order 19, and cdcdd has order 8.
Standard generators of 3.L3(7).2 are preimages C and D where CD has order 19.

### Representations

The representations of L3(7) available are
• Permutations on 57 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 152 over GF(2): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Some irreducibles in characteristic 3:
• Some irreducibles in characteristic 7:
The representations of L3(7):2 available are
• Some irreducibles in characteristic 2:
• Some irreducibles in characteristic 3:
• Some irreducibles in characteristic 7:
The representations of SL3(7) = 3.L3(7) available are
• Dimension 3 over GF(7): A and B (Meataxe), A and B (Meataxe binary), A and B (GAP). - the natural representation.
The representations of 3.L3(7):2 available are

### Maximal subgroups

The maximal subgroups of L3(7) are as follows.
• 7^2:2.L2(7):2.
• 7^2:2.L2(7):2.
• L2(7):2.
• L2(7):2.
• L2(7):2.
• (3 x A4)2.
• 3^2:Q8.
• 19:3
The maximal subgroups of L3(7):2 are as follows.
• L3(7), with generators here.
• 7^1+2:(3 x D8), with generators here.
• 2.(2 x L2(7)).2
• L2(7):2 x2.
• S3 x S4.
• 3^2:SD16.
• 19:6.

### Conjugacy classes

A set of generators for the maximal cyclic subgroups of L3(7) 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 L3(7):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 L3(7) page - ATLAS version 1.
Anonymous ftp access is also available on for.mat.bham.ac.uk.

Version 2.0 created on 22nd March 2001.
Last updated 14.12.01 by RAW.
Information checked to Level 0 on 14.12.01 by RAW.
R.A.Wilson, R.A.Parker and J.N.Bray.