Order = 95040 = 26.33.5.11.
Mult = 2.
Out = 2.

Porting notes

Porting incomplete.

Standard generators

Standard generators of M12 are a, b where a is in class 2B, b is in class 3B and ab has order 11.

Standard generators of 2.M12 are preimages A, B where A is in class +2B and B has order 6. Alternatively: A is in class +2B and AB has order 11 or B has order 6 and AB has order 11.

Standard generators of M12:2 are c, d where c is in class 2C, d is in class 3A and cd is in class 12A. Alternatively: c is in class 2C, d is in class 3A, cd has order 12 and cdcdd has order 11.

Standard generators of 2.M12.2 are preimages C, D where D has order 3.

Black box algorithms

Finding generators

Group Algorithm File
M12
M12:2

Checking generators (semi-presentations)

Group Semi-presentation File
M12 〈〈 a, b | o(a) = 2, o(b) = 3, o(ab) = 11, o(ababab2) = 6 〉〉 Download
M12:2 〈〈 c, d | o(c) = 2, o(d) = 3, o(cd) = 12, o(cdcdcdcd2) = 6 〉〉 Download

Presentations

M12 a, b | a2 = b3 = (ab)11 = [a, b]6 = (ababab−1)6 = 1 〉 Details
2.M12 A, B | A2 = B6 = [B3, A] = (AB)11 = [A, B]6 = (ABABAB−1)6B−3 = [A, BAB]5 = 1 〉 Details
M12:2 c, d | c2 = d3 = (cd)12 = (cd)5[c, d](cd−1)3cd[c, d−1]2cdcd(cd−1)3[c, d−1] = 1 〉 Details
M12:2 c, d | c2 = d3 = (cd)12 = (cdcd[c,d−1]4)2 = ((cd)4cd−1)4 = [c, dcd]5 = 1 〉 Details

Maximal subgroups

Maximal subgroups of M12

Subgroup Order Index Programs/reps
M11 7 920 12Program: Standard generators
M11 7 920 12Program: Standard generators
A6.22 1 440 66Program: Generators
A6.22 1 440 66Program: Generators
L2(11) 660 144Program: Standard generators
32:2S4 432 220Program: Generators
32:2S4 432 220Program: Generators
2 × S5 240 396Program: Generators mapping onto standard generators
21+4:S3 192 495Program: Generators
Program: Generators
42:D12 192 495Program: Generators
A4 × S3 72 1 320Program: Generators

Maximal subgroups of M12:2

Subgroup Order Index Programs/reps
M12 Program: Standard generators
L2(11):2 Program: Generators
Program: Generators
L2(11):2 Program: Generators
Program: Generators
(2 × 2 × A5).2. Program: Generators
21+4:S3.2. Program: Generators
42:D12.2. Program: Generators
31+2:D8 Program: Generators
S4 × S3 Program: Generators
S5 Program: Generators

Conjugacy classes

Conjugacy classes of M12

Conjugacy class Centraliser order Power up Class rep(s)
1A95 040 Omitted owing to length.
2A240 6A 10A (abababb)3
2B192 4A 4B 6B 8A 8B Omitted owing to length.
3A54 6B ababbababb
3B36 6A abababbabababb
4A32 8A ababababbababbabbababababbababbabb
4B32 8B ababababbabbababbababababbabbababb
5A10 10A ababababbabbababababbabb
6A12 abababb
6B6 ababb
8A8 ababababbababbabb
8B8 ababababbabbababb
10A10 ababababbabb
11A11 11B2 ab
11B11 11A2 abab

Conjugacy classes of M12:2

Conjugacy class Centraliser order Power up Class rep(s)
1A190 080
2A480 6A 10A 4C 12A
2B384 4A 6B 8A 4B 12B 12C
3A108 6B 12B 12C
3B72 6A 6C 12A
4A32 8A
5A20 10A 10B 10C
6A24 12A
6B12 12B 12C
8A8 cdcdcddcdcdcdcdcddcdcdd
10A20 cdcdcdcddcdcdd
11A11 cdcdd
2C240 6C 10B 10C
4B48 12B 12C
4C24 12A
6C12 cdcdcddcdcddcdcdd
10B20 10C3
10C20 10B3 cdcdcdd
12A12 cd
12B12 12C5
12C12 12B5 cdcdcdcddcdcddcdd