# ATLAS: Linear group L7(2)

Order = 163849992929280 = 221.34.5.72.31.127.
Mult = 1.
Out = 2.

The following information is available for L7(2):

### Standard generators

Standard generators of L7(2) are a and b where a is in class 2A, b has order 7 (necessarily class 7E), ab has order 127 and ababababb has order 7.

Type I standard generators of L7(2):2 are c and d where c is in class 2A, d is in class 12H (centr.order 144), cd has order 14 and cdcdd has order 12.

Type II standard generators of L7(2):2 are e and f where e is in class 2D, f is in class 8E/F (centr.order 92160), ef has order ?? and efeff has order ??.

NB: Class 2A is the class of transvections (in a natural 7-dimensional representation) and class 7E is the class of 7-cycles, and is the unique class of elements of order 7 with the smallest centraliser (size 49). The centraliser orders quoted for L7(2):2 are for the L7(2):2-centralisers, not the L7(2)-centralisers (unlike the ATLAS). Class 2D is the unique class of outer involutions.

### Automorphisms

An outer automorphism of L7(2) of order 2 may be obtained by mapping (a, b) to (a, b-1).

Let u be the given automorphism. Then we may obtain standard generators of L7(2):2 as follows:
Type I generators are given by c = a and d = (ab)4u(ab)10u(ab)-3u(ab)-1.
Type II generators are given by e = u and f = uab.

We can return to L7(2) by letting a = fe[e, f]4 and b = [e, f]4. (This one is the exact inverse of the above, not merely an inverse up to automorphisms.)

### Presentations

Presentations of L7(2) and L7(2):2 on their standard generators are given below.

< a, b | a2 = b7 = [a, b]4 = [a, b2]2 = [a, b3]2 = ((ab)5b-4)15 = (ababab-2)4 = 1 >.

< c, d | c2 = d12 = (cd)14 = (cd6)4 = [c, d2]3 = (cd2cd2cd6)3 = [c, d-2cdcdcd-2]2 = [c, dcd2]2 = [c, dcd3]2 = [c, dcdcd-2cdcd] = 1 >.

< e, f | e2 = f8 = (ef4)4 = [e, f]7 = [e, f2]2 = (efef2ef4)15 = (efef2efef-2)2 = ((ef)6f-3)2 = 1, more relations? >.

Just for comparison (for now), L5(2):2 on its standard generators.

< c, d | c2 = d8 = (cd)21 = (cd4)4 = [c, d]5 = [c, d2]2 = (cdcdcdcd-2)2 = 1 >.

### Representations

The representations of L7(2) available are:
• Permutations on 127a points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Permutations on 127b points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Some irreducibles in characteristic 2 (all up to dimension 1000).
• A faithful irreducible in characteristic 0
• Dimension 126 over Z: a and b (GAP).
The representations of L7(2):2 available are:
• Permutations on 254 points: e and f (Meataxe), e and f (Meataxe binary), e and f (GAP). - imprimitive.
• All irreducibles over GF(2) in dimension less than 1000:

### Maximal subgroups

The maximal subgroups of L7(2) are as follows (roughly).
The maximal subgroups of L7(2):2 are as follows (roughly).

### Conjugacy classes

We have not yet calculated the class representatives for L7(2) [117 classes] or L7(2):2 [114 classes]. Go to main ATLAS (version 2.0) page. Go to linear groups page. There is no old L7(2) page in the ATLAS version 1. Anonymous ftp access is also available. See here for details.

Version 2.0 created on 6th June 2002.
Last updated 31.03.08 by JNB.
R.A.Wilson, R.A.Parker and J.N.Bray.