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

## Porting notes

Porting incomplete.

## Standard generators

Standard generators of G2(3) are a, b where a has order 2, b is in class 3C and ab has order 13.

Standard generators of 3.G2(3) are preimages A, B where A has order 2 and AB has order 13.

Standard generators of G2(3):2 are c, d where c has order 2 (necessarily 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, D where CD has order 13.

## Presentations

G2(3) 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 〉 Details
G2(3):2 c, d | c2 = d4 = (cd)13 = (cdcd2cd2)2 = [c, dcdcdcd−1cdcd(cd−1)3cd(cd−1)2] = 1 〉 Details

## Representations

### Representations of 3.G2(3)

• View detailed report.
• Permutation representations:
Number of points ID Generators Description Link
1134 Std Details
• Matrix representations
Char Ring Dimension ID Generators Description Link
2 GF(4) 27 Std Details
7 GF(7) 27 Std Details
13 GF(13) 27 Std Details

## Maximal subgroups

### Maximal subgroups of G2(3)

Subgroup Order Index Programs/reps
U3(3):2 Program: Generators
U3(3):2 Program: Generators
(32 × 31+2):2S4 Program: Generators
(32 × 31+2):2S4 Program: Generators
L3(3):2 Program: Generators
L3(3):2 Program: Generators
L2(8):3 Program: Generators
23.L3(2) Program: Generators
L2(13) Program: Generators
21+4:32:2 Program: Generators

## Conjugacy classes

### Conjugacy classes of G2(3)

Conjugacy class Centraliser order Power up Class rep(s)
1A 4 245 696 abababbabababbabababbabababbabababbabababbabababbabababb
2A 576 abababbabababbabababbabababb
3A 5 832 ababababbabababababbabababababbabababababbab
3B 5 832 Omitted owing to length.
3C 729 (babababbab)3
3D 162 abababbabababbabababababbabababbabab
3E 162 aabababbabababbabaabababbabababbab
4A 96 abababbabababb
4B 96 aabababbabababbababbaabababbabababbababb
6A 72 ababababbabababababbab
6B 72 abababbabababbababababbabababbab
6C 18 abababbabababbabab
6D 18 aabababbabababbab
7A 7 abababbab
8A 8 abababb
8B 8 aabababbabababbababb
9A 27 babababbab
9B 27 abbababababbabbabababbababbababababbabbabababbab
9C 27 abbababababbabbabababbab
12A 12 ababababbab
12B 12 abababbabababbab
13A 13 ab
13B 13 abab

### Conjugacy classes of G2(3):2

Conjugacy class Centraliser order Power up Class rep(s)
1A 8 491 392 Omitted owing to length.
2A 1 152 dcdcdcdddcdcdcdddcdcdcdddcdcdcdd
3A 5 832 Omitted owing to length.
3C 1 458 (cdcddcdcdd)3
3D 324 cdddcdcdcddcdddcdcdcddcdddcdcdcddcdddcdcdcdd
3E 324 cdcdcdcddcdddcdcdcddcdcdcdcddcdddcdcdcdd
4A 96 dcdcdcdddcdcdcdd
6A 72 Omitted owing to length.
6C 36 cdddcdcdcddcdddcdcdcdd
6D 36 cdcdcdcddcdddcdcdcdd
7A 14 cdcdcddcdcdcdd
8A 8 dcdcdcdd
9A 54 cdcddcdcdd
9B 54 cdddcdcdcddcdcdcddcdddcdcdcddcdddcdcdcddcdcdcddcdddcdcdcdd
9C 54 Omitted owing to length.
12A 12 cdcddcdddcdcdcddcdcdcddcdddcdcdcdd
13A 13 cd
2B 3 024 (ccdcddcdddcdcdcddcdcdcddcdddcdcdcdd)3
4C 48 (cdddcdcdcdd)3
6E 54 (cdcdd)3
6G 36 (ccdcddcdddcdcdcddcdcdcddcdddcdcdcdd)5
6F 36 ccdcddcdddcdcdcddcdcdcddcdddcdcdcdd
12C 12 (cdddcdcdcdd)7
12D 12 cdddcdcdcdd
14A 14 cdcdcdd
18A 18 cdcdd
18B 18 (cdddcdcdcddcdcdcddcdddcdcdcdd)5
18C 18 cdddcdcdcddcdcdcddcdddcdcdcdd