# ATLAS: Exceptional group G2(3)

Order = 4245696 = 26.36.7.13.
Mult = 3.
Out = 2.

### Standard generators

Standard generators of G2(3) are a and b where a has order 2, b is in class 3C and ab has order 13.
Standard generators of the triple cover 3.G2(3) are preimages A and B where A has order 2 and AB has order 13.

Standard generators of G2(3):2 are c and d where c has order 2 (and is in class 2B), d is in class 4C, cd has order 13 and cdd has order 6.
Standard generators of 3.G2(3):2 are preimages C and D where CD has order 13.

### Automorphisms

The outer automorphism of G2(3) can be realised by mapping (a, b) to (a, (abb)-3b(abb)3).

### Presentations

Presentations of G2(3) and G2(3):2 on their standard generators are given below.

< a, b | a2 = b3 = (ab)13 = [a, b]13 = abab[a, b]4(ab)3[a, bab]3 = (((ab)3ab-1)2(ab)2(ab-1)2)2 = 1 >.

< c, d | c2 = d4 = (cd)13 = (cdcd2cd2)2 = [c, dcdcdcd-1cdcd(cd-1)3cd(cd-1)2] = 1 >.

### Representations

The representations of G2(3) available are:
• Some faithful irreducible representaions in characteristic 2.
• All faithful irreducibles in characteristic 3.
• Dimension 7 over GF(3) - the natural representation: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 7 over GF(3) - the image of the above under an outer automorphism: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 27 over GF(3): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 27 over GF(3): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 49 over GF(3): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 189 over GF(3): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 189 over GF(3): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 729 over GF(3) - the Steinberg representation: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• a and b as 14 × 14 matrices over Z.
The representations of 3.G2(3) available are:
The representations of G2(3):2 available are:
The representation of 3.G2(3):2 available is:

### Maximal subgroups

The maximal subgroups of G2(3) are:
The maximal subgroups of G2(3):2 are:

### Conjugacy classes

A set of generators for the maximal cyclic subgroups of G2(3) 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(3):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.

An outer automorphism of G2(3) can be obtained by running this program on the standard generators.

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