# ATLAS: Exceptional group G2(4)

Order = 251596800 = 212.33.52.7.13.
Mult = 2.
Out = 2.

### Standard generators

Standard generators of G2(4) are a and b where a is in class 2A, b is in class 5C/D, ab has order 13 and abb has order 13.
To ensure that b is in class 5C/D (and not class 5A/B), also check that ababb has order 15.
The representations below are labelled such that b is in class 5C of the Atlas of Brauer Characters.
Standard generators of the double cover 2.G2(4) are preimages A and B where B has order 5 and AB has order 13.

Standard generators of G2(4):2 are c and d where c is in class 2C, d is in class 4D, cd has order 13 and cdcddd has order 6.
Standard generators of the Atlas double cover 2.G2(4):2 are preimages C and D where CD has order 13.
(Similarly for the isoclinic, non­Atlas, double cover 2.G2(4).2.)

### Automorphisms

The outer automorphism of G2(4) can be realised by the map taking (a, b) to (a, b2).

### Black box algorithms

To find standard generators of G2(4):
• Find an element x of order 2 by taking a suitable power of any element of order 4, 8 or 12.
[The probability of success at each attempt is 33 in 256 (about 1 in 8).]
• Find an element y of order 5 by taking a suitable power of any element of order 5, 10 or 15.
[The probability of success at each attempt is 12 in 25 (about 1 in 2), and the porbability that y is in the right class is 1 in 2.]
• Find conjugates a of x and b of y such that ab has order 13 and abb has order 13.
[The probability of success at each attempt is 20 in 273 (about 1 in 14).]
• If ababb has order 15, then y was in the right class and a and b are standard generators of G2(4). Otherwise ababb has order 10 and y was in the wrong class, so go back to Step 2.
To find standard generators of G2(4).2:
• Find any element of order 14. Its seventh power, x say, is in class 2C.
[The probability of success at each attempt is 1 in 14 (or 1 in 7 if we search through outer elements only).]
• Er, I don't know how to find a 4D-element. 15/64 of the outer elements have order 4 or 12, and among these just 1/5 power into class 4D.
• Suppose we have obtained y in class 4D.
• Find conjugates c of x and d of y such that cd has order 13 and cdcddd has order 6.
[The probability of success at each attempt is 6 in 325 (about 1 in 54).]
• Now c and d are standard generators of G2(4).2.
Alternatively, to find standard generators of G2(4).2:
• Find any element of order 14. Its seventh power, x say, is in class 2C.
[The probability of success at each attempt is 1 in 14 (or 1 in 7 if we search through outer elements only).]
• Find an element y of order 13.
[The probability of success at each attempt is 1 in 13 (or 2 in 13 if we search through inner elements only).]
• Find conjugates c of x and z of y such that cz has order 4, czz has order 6, czzz has order 6.
[The probability of success at each attempt is 1 in 1600.]
• Now c and d = cz are standard generators of G2(4).2.

### Representations

The representations of G2(4) available are:
• Some representations in characteristic 2:
• Some representations in characteristic 3.
• Some representations in characteristic 5.
• Dimension 65 over GF(7): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 78 over GF(7): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 65 over GF(13): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 78 over GF(13): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Some faithful irreducibles in characteristic 0
• Dimension 65 over Z: a and b (GAP).
• Dimension 78 over Z: a and b (GAP).
• Dimension 350 over Z: a and b (GAP).
• Some primitive permutation representations.
• Permutations on 416 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Permutations on 1365[a] points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Permutations on 1365[b] points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Permutations on 2016 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Permutations on 2080 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Permutations on 20800 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
The representations of 2.G2(4) available are:
• Dimension 12 over GF(3): A and B (Meataxe), A and B (Meataxe binary), A and B (GAP).
• Dimension 12 over GF(5): A and B (Meataxe), A and B (Meataxe binary), A and B (GAP).
• Dimension 92 over GF(5): A and B (Meataxe), A and B (Meataxe binary), A and B (GAP).
• Dimension 12 over GF(7): A and B (Meataxe), A and B (Meataxe binary), A and B (GAP).
• Dimension 12 over GF(13): A and B (Meataxe), A and B (Meataxe binary), A and B (GAP).
• Some faithful irreducibles in characteristic 0
• Dimension 12 over Z[i]: A and B (GAP).
The representations of G2(4):2 available are:
The representations of 2.G2(4):2 available are:
The representations of the non­split 2.G2(4).2 available are:

### Maximal subgroups

The maximal subgroups of G2(4) are:
• J2, with standard generators here.
• 22+8:(A5 × 3), with generators here (mapping to standard generators of A5).
• 24+6:(A5 × 3), with generators here.
• U3(4):2, with standard generators here.
• 3.L3(4):2, with (non­standard) generators here.
• U3(3):2, with (non­standard) generators here.
• A5 × A5, with generators here.
• L2(13), with standard generators here.
The maximal subgroups of G2(4):2 are:

### Conjugacy classes

A set of generators for the maximal cyclic subgroups of G2(4) 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 G2(4):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 exceptional groups page. Go to old G2(4) page - ATLAS version 1. Anonymous ftp access is also available. See here for details.

Version 2.0 created on 14th April 2000.
Last updated 02.03.11 by JNB.
Information checked to Level 0 on 14.04.00 by RAW.
R.A.Wilson, R.A.Parker and J.N.Bray.