# ATLAS: Exceptional group G2(5)

Order = 5859000000 = 26.33.56.7.31.
Mult = 1.
Out = 1.

### Standard generators

Type I standard generators of G2(5) are a and b where a has order 2 (is in class 2A), b is in class 3B, ab has order 7 and ababb has order 15.
Type II standard generators of G2(5) are x and y where x has order 2 (is in class 2A), y is in class 5B and xy has order 7.

Without loss of generality, we can obtain (a, b) as (a, b) = (x, ((xyxyxy4)4)(xy2)23(xy)5).
Conversely the pair (X, Y) = (a, (babab)3) is conjugate (in G2(5)) to (x, y).
In fact, (x, y) = (X, Yc) = (Xc, Yc), where c = de4dede3dede2d, with d = [X, YXY2]6 and e = YXYXY[X, YXYXY]7.

### Representations

The representations of G2(5) available are:
• Primitive permutation representations.
• Permutations on 3906 points - on the cosets of N(5A): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Permutations on 3906 points - on the cosets of N(5A2): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Permutations on 7750 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Permutations on 7875 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 124 over GF(2): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 124 over GF(3): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 651 over GF(3): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Faithful irreducibles in characteristic 5 (up to dimension 1000).

### Maximal subgroups

The maximal subgroups of G2(5) are as follows.

### Conjugacy classes

A set of generators for the maximal cyclic subgroups 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 exceptional groups page. Go to old G2(5) page - ATLAS version 1. Anonymous ftp access is also available on for.mat.bham.ac.uk.

Version 2.0 created on 12th May 1999.
Last updated 03.12.08 by JNB.
Information checked to Level 0 on 12.05.99 by JNB.
R.A.Wilson, R.A.Parker and J.N.Bray.