# 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 irreducibles in characteristic 2.
• a and b as 14 × 14 matrices over GF(2).
• a and b as 64 × 64 matrices over GF(4).
• a and b as 78 × 78 matrices over GF(2).
• a and b as 90 × 90 matrices over GF(2).
• a and b as 90 × 90 matrices over GF(2).
• a and b as 90 × 90 matrices over GF(2).
• a and b as 378 × 378 matrices over GF(2).
• All faithful irreducibles in characteristic 3.
• a and b as 7 × 7 matrices over GF(3) - the natural representation.
• a and b as 7 × 7 matrices over GF(3) - the image of the above under an outer automorphism.
• a and b as 27 × 27 matrices over GF(3).
• a and b as 27 × 27 matrices over GF(3).
• a and b as 49 × 49 matrices over GF(3).
• a and b as 189 × 189 matrices over GF(3).
• a and b as 189 × 189 matrices over GF(3).
• a and b as 729 × 729 matrices over GF(3) - the Steinberg representation.
• a and b as 14 × 14 matrices over Z.
The representations of 3.G2(3) available are:
• A and B as 27 × 27 matrices over GF(4).
• A and B as 27 × 27 matrices over GF(7).
• A and B as 27 × 27 matrices over GF(13).
• A and B as permutations on 1134 points.
The representations of G2(3):2 available are:
• c and d as permutations on 756 points.
• c and d as 14 × 14 matrices over GF(2).
• c and d as 14 × 14 matrices over GF(3).
• c and d as 14 × 14 matrices over GF(7).
• c and d as 14 × 14 matrices over GF(13).
• c and d as 14 × 14 matrices over Z[i3].
The representation of 3.G2(3):2 available is:
• C and D as 54 × 54 matrices over GF(2).

### Maximal subgroups

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