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

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
J2
Download
J2:2
Download

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

Group Presentation Link
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

Representations

Representations of J2

Representations of 2.J2

Representations of J2:2

Representations of 2.J2.2

Maximal subgroups

Maximal subgroups of J2

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

Maximal subgroups of J2:2

Subgroup Order Index Programs/reps
J2 604 800 2Program: Standard generators
U3(3):2 12 096 100Program: Standard generators
Program: Generators
3.A6.22 4 320 280Program: Generators
Program: Generators
21+4.S5 3 840 315Program: Generators
22+4:(3 × S3).2 2 304 525Program: Generators
(A4 × A5):2 1 140 1 061Program: Generators
(A5 × D10).2 1 200 1 008Program: Generators
L3(2):2 × 2 672 1 800Program: Generators mapping onto standard generators
Program: Generators
52:(4 × S3) 600 2 016Program: Generators
S5 120 10 080Program: 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)
1A604 800 Omitted owing to length.
2A1 920 4A 6A 8A 10C 10D 12A Omitted owing to length.
2B240 6B 10A 10B (abababbababbababb)3
3A1 080 6A 12A 15A 15B ababbababbababbababb
3B36 6B abababbababbababbabababbababbababb
4A96 8A 12A Omitted owing to length.
5A300 5B2 10A 10B 15A 15B Omitted owing to length.
5B300 5A2 10A 10B 15A 15B Omitted owing to length.
5C50 5D2 10C 10D abababbababbabbabababbababbabbabababbababbabbabababbababbabb
5D50 5C2 10C 10D abababbababbabbabababbababbabb
6A24 12A ababbababb
6B12 abababbababbababb
7A7 ab
8A8 bbabababbababbabbabababbababbababb
10A20 10B3 (abababbababbabbabababbababbababb)3
10B20 10A3 abababbababbabbabababbababbababb
10C10 10D3 abababbababbabb
10D10 10C3 (abababbababbabb)3
12A12 ababb
15A15 15B2 babababbababbabbabababbababbababb
15B15 15A2 Omitted owing to length.

Download words for class representatives.

Conjugacy classes of J2:2

Conjugacy class Centraliser order Power up Class rep(s)
1A1 209 600
2A3 840 4A 6A 8A 10C 12A 4B 8B 8C 12B 24A 24B
2B480 6B 10A 4C 12C
3A2 160 6A 12A 15A 12B 24A 24B
3B72 6B 6C 12C
4A192 8A 12A 8B 8C 24A 24B
5A300 10A 15A
5B50 10C
6A48 12A 12B 24A 24B
6B24 12C
7A14 14A
8A16 cdcdcddcdcdcdcdcdd
10A20 cdcdcdddcdcdcddcdcdd
10C10 cdcdd
12A24 24A 24B
15A15 cdcdcddcd
2C672 6C 14A
4B96 12B
4C24 12C
6C12 cdcdcddcdcd
8B96 24A 24B
8C32 dcdcdcddcdcdd
12B12 cdcdcddcdcdd
12C12 cdcdcdd
14A14 cd
24A24 24B7 cdd
24B24 24A7

Download words for class representatives.