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)
1A251 596 800 Omitted owing to length.
2A61 440 Omitted owing to length.
2B3 840 (babbabbababb)3
3A60 480 Omitted owing to length.
3B180 babbabbababbbabbabbababb
4A1 536 Omitted owing to length.
4B768 (abbababbabababbabbababb)3
4C512 abbababbabbababb
5A300 aabababbabbababbaabababbabbababb
5B300 Omitted owing to length.
5C300 abababbabbababbabababbabbababb
5D300 abababbabbababbabababbabbababbabababbabbababbabababbabbababb
6A192 ababbabbababbabbababbababbabbababbabbababb
6B12 babbabbababb
7A21 (abababb)3
8A32 babababbabbababbaabababbabbababb
8B32 abbababb
10A20 (abababbabbababb)3
10B20 abababbabbababb
10C20 (aabababbabbababb)3
10D20 aabababbabbababb
12A48 ababbabbababbabbababb
12B48 abbababbabababbabbababb
12C48 (abbababbabababbabbababb)5
13A13 abab
13B13 ab
15A15 ababb
15B15 ababbababb
15C15 abbabbababbabbabbababb
15D15 abbabbababb
21A21 abababb
21B21 abababbabababb

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Conjugacy classes of G2(4):2

Conjugacy class Centraliser order Power up Class rep(s)
1A503 193 600 (cdddcdcdcddcdcdcdddcdcdcdd)8
2A122 880 (cdddcdcdcddcdcdcdddcdcdcdd)4
2B7 680 (cdcdcdcdd)6
3A120 960 (ccdcdcdddcdcdcddcdcdd)2
3B360 (cdd)2
4A3 072 (cdddcdcdcddcdcdcdddcdcdcdd)2
4B1 536 (cddcdcdcddcdcdcdcdd)6
4C1 024 (cdddcdcdcdd)4
5A300 (cdcdcddcdcdcdcdd)2
5C300 (cdcdcdddcdcdcddcdcdd)2
6A384 (dcdcdcddcdddcdcdcdd)2
6B24 (cdcdcdcdd)2
7A42 (cdcdcdddcdcdcdd)2
8A64 cdddcdcdcddcdcdcdddcdcdcdd
8B64 (cdddcdcdcdd)2
10A20 cdcdcdddcdcdcddcdcdd
10C20 cdcdcddcdcdcdcdd
12A96 (cdcdcdd)2
12B96 (cddcdcdcddcdcdcdcdd)2
12C96 (cddcdcdcddcdcdcdcdd)10
13A13 cd
15A15 dcdcdcdcdd
15C15 cdcdcddcdddcdcdcdd
21A21 dcdcdcdd
2C24 192 (cdd)3
4D384 (dcdcdcddcdddcdcdcdd)3
4E96 (cdcdcdcdd)3
6C432 ccdcdcdddcdcdcddcdcdd
6D36 cdd
8C192 (cdcdcdd)3
8D96 (cddcdcdcddcdcdcdcdd)3
8E64 ccdddcdcdcddcdcdcdddcdcdcdd
12D48 dcdcdcddcdddcdcdcdd
12E12 cdcdcdcdd
14A14 cdcdcdddcdcdcdd
16A16 cdddcdcdcdd
16B16 (cdddcdcdcdd)3
24A24 cdcdcdd
24B24 cddcdcdcddcdcdcdcdd
24C24 (cddcdcdcddcdcdcdcdd)5

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