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
1134StdDetails
• Matrix representations
Char Ring Dimension ID Generators Description Link
2GF(4)27StdDetails
7GF(7)27StdDetails
13GF(13)27StdDetails

## 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)
1A4 245 696 abababbabababbabababbabababbabababbabababbabababbabababb
2A576 abababbabababbabababbabababb
3A5 832 ababababbabababababbabababababbabababababbab
3B5 832 Omitted owing to length.
3C729 (babababbab)3
3D162 abababbabababbabababababbabababbabab
3E162 aabababbabababbabaabababbabababbab
4A96 abababbabababb
4B96 aabababbabababbababbaabababbabababbababb
6A72 ababababbabababababbab
6B72 abababbabababbababababbabababbab
6C18 abababbabababbabab
6D18 aabababbabababbab
7A7 abababbab
8A8 abababb
8B8 aabababbabababbababb
9A27 babababbab
9B27 abbababababbabbabababbababbababababbabbabababbab
9C27 abbababababbabbabababbab
12A12 ababababbab
12B12 abababbabababbab
13A13 ab
13B13 abab

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

Conjugacy class Centraliser order Power up Class rep(s)
1A8 491 392 Omitted owing to length.
2A1 152 dcdcdcdddcdcdcdddcdcdcdddcdcdcdd
3A5 832 Omitted owing to length.
3C1 458 (cdcddcdcdd)3
3D324 cdddcdcdcddcdddcdcdcddcdddcdcdcddcdddcdcdcdd
3E324 cdcdcdcddcdddcdcdcddcdcdcdcddcdddcdcdcdd
4A96 dcdcdcdddcdcdcdd
6A72 Omitted owing to length.
6C36 cdddcdcdcddcdddcdcdcdd
6D36 cdcdcdcddcdddcdcdcdd
7A14 cdcdcddcdcdcdd
8A8 dcdcdcdd
9A54 cdcddcdcdd
9B54 cdddcdcdcddcdcdcddcdddcdcdcddcdddcdcdcddcdcdcddcdddcdcdcdd
9C54 Omitted owing to length.
12A12 cdcddcdddcdcdcddcdcdcddcdddcdcdcdd
13A13 cd
2B3 024 (ccdcddcdddcdcdcddcdcdcddcdddcdcdcdd)3
4C48 (cdddcdcdcdd)3
6E54 (cdcdd)3
6G36 (ccdcddcdddcdcdcddcdcdcddcdddcdcdcdd)5
6F36 ccdcddcdddcdcdcddcdcdcddcdddcdcdcdd
12C12 (cdddcdcdcdd)7
12D12 cdddcdcdcdd
14A14 cdcdcdd
18A18 cdcdd
18B18 (cdddcdcdcddcdcdcddcdddcdcdcdd)5
18C18 cdddcdcdcddcdcdcddcdddcdcdcdd