Order = 604800 = 27.33.52.7.
Mult = 2.
Out = 2.

## Porting notes

Fully copied from version 2. 1/8/06.

## Standard generators

Standard generators of J2 = HJ are a, b where a is in class 2B, b is in class 3B, ab has order 7 and ababb has order 12.

Standard generators of 2.J2 are preimages A, B where B has order 3 and AB has order 7.

Standard generators of J2:2 are c, d where c is in class 2C, d is in class 5AB and cd has order 14.

Standard generators of 2.J2.2 are preimages C, D where D has order 5.

Standard generators of 2.J2:2 (isoclinic) are preimages C, D where D has order 5.

### Notes

• (2.J2.2) A pair of generators conjugate to A, B can be obtained as
A' = (CDCDCDD)18,
B' = (CDD)−3(CDCDCDD)16(CDD)3.

## Automorphisms

Group Automorphism Order in Out(G) Description
J2 = HJ Outer automorphism 2 a maps to a
b maps to bb
(No program)
2.J2 Outer automorphism 2 A maps to A−1
B maps to BB
(No program)

(J2) Outer automorphism is in class 2C.

## Black box algorithms

### Finding generators

Group Algorithm File

### Checking generators (semi-presentations)

Group Semi-presentation File
J2 〈〈 a, b | o(a) = 2, o(b) = 3, o(ab) = 7, o(abab2) = 12 〉〉 Download
J2:2 〈〈 c, d | o(c) = 2, o(d) = 5, o(cd) = 14, o(cdd) = 24 〉〉 Download

## Presentations

J2 a, b | a2 = b3 = (ab)7 = [a, b]12 = (ababab−1abab−1ab−1ababab−1ab−1abab−1)3 = 1 〉 Details
J2 a, b | a2 = b3 = (ab)7 = [a, b]12 = (ababab−1abab−1)6 = 1 〉 Details
2.J2 A, B | A4 = [A2,B] = B3 = (AB)7 = [A,B]12 = (ABABAB−1ABAB−1)6 = A−2(ABABAB−1ABAB−1AB−1ABABAB−1AB−1ABAB−1)3 = 1 〉 Details
2.J2 A, B | A4 = [A2,B] = B3 = (AB)7 = [A,B]12 = (ABABAB−1ABAB−1)6 = A−2(ABABAB−1ABAB−1AB−1ABABAB−1AB−1ABAB−1)3 = 1 〉 Details
J2:2 c, d | c2 = d5 = (cd)14 = [c, d]7 = (cdcdcd−2cd−2)3 = [c, dcd]3 = (cdcdcd2)3cd−1cdcdcd−1cd2 = 1 〉 Details
2.J2.2 C, D | C4 = [C2,D] = D5 = C−2(CD)14 = [C,D]7 = [D,CDCD−2CD−1CD2CDC] = C−2(CDCDCD−2CD−2)3 = C−2(CDCDCD−2)4 = C−2[C,DCD]3 = C−2(CDCDCD2)3CD−1CDCDCD−1CD2 = C−2[C,DCD2]3 = 1 〉 Details
2.J2:2 (isoclinic) C, D | C2 = D5 = (CD)14 = [C,D]7 = [D,CDCD2CD−1CD2CDC] = (CDCDCD−2CD−2)3[C,DCD]3 = (CDCDCD−2)4 = (CDCDCD2)3CD−1CDCDCD−1CD2 = [C,DCD2]3 = 1 〉 Details

## Maximal subgroups

### Maximal subgroups of J2

Subgroup Order Index Programs/reps
U3(3) 6 048 100 Program: Generators
3.A6.2 2 160 280 Program: Generators
21+4:A5 1 920 315 Program: Generators
22+4:(3 × S3) 1 152 525 Program: Generators
A4 × A5 720 840 Program: Generators
A5 × D10 600 1 008 Program: Generators
L3(2):2 336 1 800 Program: Generators
52:D12 300 2 016 Program: Generators
A5 60 10 080 Program: Generators

### Maximal subgroups of J2:2

Subgroup Order Index Programs/reps
J2 604 800 2 Program: Standard generators
U3(3):2 12 096 100 Program: Standard generators
Program: Generators
3.A6.22 4 320 280 Program: Generators
Program: Generators
21+4.S5 3 840 315 Program: Generators
22+4:(3 × S3).2 2 304 525 Program: Generators
(A4 × A5):2 1 140 1 061 Program: Generators
(A5 × D10).2 1 200 1 008 Program: Generators
L3(2):2 × 2 672 1 800 Program: Generators mapping onto standard generators
Program: Generators
52:(4 × S3) 600 2 016 Program: Generators
S5 120 10 080 Program: Standard generators
Program: Generators

### Notes

(J2) Words for maximal subgroups provided by Peter Walsh, implemented and checked by Ibrahim Suleiman.

## Conjugacy classes

### Conjugacy classes of J2

Conjugacy class Centraliser order Power up Class rep(s)
1A 604 800 Omitted owing to length.
2A 1 920 4A 6A 8A 10C 10D 12A Omitted owing to length.
2B 240 6B 10A 10B (abababbababbababb)3
3A 1 080 6A 12A 15A 15B ababbababbababbababb
3B 36 6B abababbababbababbabababbababbababb
4A 96 8A 12A Omitted owing to length.
5A 300 5B2 10A 10B 15A 15B Omitted owing to length.
5B 300 5A2 10A 10B 15A 15B Omitted owing to length.
5C 50 5D2 10C 10D abababbababbabbabababbababbabbabababbababbabbabababbababbabb
5D 50 5C2 10C 10D abababbababbabbabababbababbabb
6A 24 12A ababbababb
6B 12 abababbababbababb
7A 7 ab
8A 8 bbabababbababbabbabababbababbababb
10A 20 10B3 (abababbababbabbabababbababbababb)3
10B 20 10A3 abababbababbabbabababbababbababb
10C 10 10D3 abababbababbabb
10D 10 10C3 (abababbababbabb)3
12A 12 ababb
15A 15 15B2 babababbababbabbabababbababbababb
15B 15 15A2 Omitted owing to length.

### Conjugacy classes of J2:2

Conjugacy class Centraliser order Power up Class rep(s)
1A 1 209 600
2A 3 840 4A 6A 8A 10C 12A 4B 8B 8C 12B 24A 24B
2B 480 6B 10A 4C 12C
3A 2 160 6A 12A 15A 12B 24A 24B
3B 72 6B 6C 12C
4A 192 8A 12A 8B 8C 24A 24B
5A 300 10A 15A
5B 50 10C
6A 48 12A 12B 24A 24B
6B 24 12C
7A 14 14A
8A 16 cdcdcddcdcdcdcdcdd
10A 20 cdcdcdddcdcdcddcdcdd
10C 10 cdcdd
12A 24 24A 24B
15A 15 cdcdcddcd
2C 672 6C 14A
4B 96 12B
4C 24 12C
6C 12 cdcdcddcdcd
8B 96 24A 24B
8C 32 dcdcdcddcdcdd
12B 12 cdcdcddcdcdd
12C 12 cdcdcdd
14A 14 cd
24A 24 24B7 cdd
24B 24 24A7