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 Download
M12:2 Download

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

Group Presentation Link
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

Representations

Representations of M12

Representations of 2.M12

Representations of M12:2

Representations of 2.M12.2

Maximal subgroups

Maximal subgroups of M12

Subgroup Order Index Programs/reps
M11 7 920 12 Program: Standard generators
M11 7 920 12 Program: Standard generators
A6.22 1 440 66 Program: Generators
A6.22 1 440 66 Program: Generators
L2(11) 660 144 Program: Standard generators
32:2S4 432 220 Program: Generators
32:2S4 432 220 Program: Generators
2 × S5 240 396 Program: Generators mapping onto standard generators
21+4:S3 192 495 Program: Generators
Program: Generators
42:D12 192 495 Program: Generators
A4 × S3 72 1 320 Program: 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)
1A 95 040 Omitted owing to length.
2A 240 6A 10A (abababb)3
2B 192 4A 4B 6B 8A 8B Omitted owing to length.
3A 54 6B ababbababb
3B 36 6A abababbabababb
4A 32 8A ababababbababbabbababababbababbabb
4B 32 8B ababababbabbababbababababbabbababb
5A 10 10A ababababbabbababababbabb
6A 12 abababb
6B 6 ababb
8A 8 ababababbababbabb
8B 8 ababababbabbababb
10A 10 ababababbabb
11A 11 11B2 ab
11B 11 11A2 abab

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Conjugacy classes of M12:2

Conjugacy class Centraliser order Power up Class rep(s)
1A 190 080
2A 480 6A 10A 4C 12A
2B 384 4A 6B 8A 4B 12B 12C
3A 108 6B 12B 12C
3B 72 6A 6C 12A
4A 32 8A
5A 20 10A 10B 10C
6A 24 12A
6B 12 12B 12C
8A 8 cdcdcddcdcdcdcdcddcdcdd
10A 20 cdcdcdcddcdcdd
11A 11 cdcdd
2C 240 6C 10B 10C
4B 48 12B 12C
4C 24 12A
6C 12 cdcdcddcdcddcdcdd
10B 20 10C3
10C 20 10B3 cdcdcdd
12A 12 cd
12B 12 12C5
12C 12 12B5 cdcdcdcddcdcddcdd

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