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

Porting notes

Porting incomplete.

Standard generators

Standard generators of G2(4) are a, b where a is in class 2A, b is in class 5C/D, ab has order 13 and abb has order 13. Alternatively: a is in class 2A, b has order 5, ab has order 13, abb has order 13 and ababb has order 15.

Standard generators of 2.G2(4) are preimages A, B where B has order 5 and AB has order 13.

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

Standard generators of 2.G2(4).2 are preimages C, D where CD has order 13.

Standard generators of 2.G2(4).2 (isoclinic) are preimages C, D where CD has order 13.

Representations

Representations of G2(4)

Representations of 2.G2(4)

Representations of G2(4):2

Representations of 2.G2(4).2

Representations of 2.G2(4).2 (isoclinic)

Maximal subgroups

Maximal subgroups of G2(4)

Subgroup Order Index Programs/reps
J2 Program: Standard generators
22+8:(A5 × 3) Program: Generators mapping onto standard generators
24+6:(A5 × 3) Program: Generators
U3(4):2 Program: Generators
3.L3(4):2 Program: Generators
U3(3):2 Program: Generators
A5 × A5 Program: Generators
L2(13) Program: Standard generators

Maximal subgroups of G2(4):2

Subgroup Order Index Programs/reps
G2(4) Program: Standard generators
J2:2
22+8:(A5 × 3):2
24+6:(A5 × 3):2
U3(4):4
3.L3(4).2.2
U3(3):2 × 2
(A5 × A5):2
L2(13):2

Conjugacy classes

Conjugacy classes of G2(4)

Conjugacy class Centraliser order Power up Class rep(s)
1A 251 596 800 Omitted owing to length.
2A 61 440 Omitted owing to length.
2B 3 840 (babbabbababb)3
3A 60 480 Omitted owing to length.
3B 180 babbabbababbbabbabbababb
4A 1 536 Omitted owing to length.
4B 768 (abbababbabababbabbababb)3
4C 512 abbababbabbababb
5A 300 aabababbabbababbaabababbabbababb
5B 300 Omitted owing to length.
5C 300 abababbabbababbabababbabbababb
5D 300 abababbabbababbabababbabbababbabababbabbababbabababbabbababb
6A 192 ababbabbababbabbababbababbabbababbabbababb
6B 12 babbabbababb
7A 21 (abababb)3
8A 32 babababbabbababbaabababbabbababb
8B 32 abbababb
10A 20 (abababbabbababb)3
10B 20 abababbabbababb
10C 20 (aabababbabbababb)3
10D 20 aabababbabbababb
12A 48 ababbabbababbabbababb
12B 48 abbababbabababbabbababb
12C 48 (abbababbabababbabbababb)5
13A 13 abab
13B 13 ab
15A 15 ababb
15B 15 ababbababb
15C 15 abbabbababbabbabbababb
15D 15 abbabbababb
21A 21 abababb
21B 21 abababbabababb

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Conjugacy classes of G2(4):2

Conjugacy class Centraliser order Power up Class rep(s)
1A 503 193 600 (cdddcdcdcddcdcdcdddcdcdcdd)8
2A 122 880 (cdddcdcdcddcdcdcdddcdcdcdd)4
2B 7 680 (cdcdcdcdd)6
3A 120 960 (ccdcdcdddcdcdcddcdcdd)2
3B 360 (cdd)2
4A 3 072 (cdddcdcdcddcdcdcdddcdcdcdd)2
4B 1 536 (cddcdcdcddcdcdcdcdd)6
4C 1 024 (cdddcdcdcdd)4
5A 300 (cdcdcddcdcdcdcdd)2
5C 300 (cdcdcdddcdcdcddcdcdd)2
6A 384 (dcdcdcddcdddcdcdcdd)2
6B 24 (cdcdcdcdd)2
7A 42 (cdcdcdddcdcdcdd)2
8A 64 cdddcdcdcddcdcdcdddcdcdcdd
8B 64 (cdddcdcdcdd)2
10A 20 cdcdcdddcdcdcddcdcdd
10C 20 cdcdcddcdcdcdcdd
12A 96 (cdcdcdd)2
12B 96 (cddcdcdcddcdcdcdcdd)2
12C 96 (cddcdcdcddcdcdcdcdd)10
13A 13 cd
15A 15 dcdcdcdcdd
15C 15 cdcdcddcdddcdcdcdd
21A 21 dcdcdcdd
2C 24 192 (cdd)3
4D 384 (dcdcdcddcdddcdcdcdd)3
4E 96 (cdcdcdcdd)3
6C 432 ccdcdcdddcdcdcddcdcdd
6D 36 cdd
8C 192 (cdcdcdd)3
8D 96 (cddcdcdcddcdcdcdcdd)3
8E 64 ccdddcdcdcddcdcdcdddcdcdcdd
12D 48 dcdcdcddcdddcdcdcdd
12E 12 cdcdcdcdd
14A 14 cdcdcdddcdcdcdd
16A 16 cdddcdcdcdd
16B 16 (cdddcdcdcdd)3
24A 24 cdcdcdd
24B 24 cddcdcdcddcdcdcdcdd
24C 24 (cddcdcdcddcdcdcdcdd)5

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