Order = 898128000 = 27.36.53.7.11.
Mult = 3.
Out = 2.

Porting notes

Porting incomplete.

Standard generators

Standard generators of McL are a, b where a is in class 2A, b is in class 5A, ab has order 11 and ababababbababbabb has order 7.

Standard generators of 3.McL are preimages A, B where A has order 2 and B has order 5.

Standard generators of McL:2 are c, d where c is in class 2B, d is in class 3B, cd has order 22 and cdcdcdcddcdcddcdd has order 24.

Black box algorithms

Finding generators

Group Algorithm File
McL Download
McL:2 Download

Checking generators (semi-presentations)

Group Semi-presentation File
McL 〈〈 a, b | o(a) = 2, o(b) = 5, o(ab) = 11, o(ababababbababbabb) = 7, o(ab2) = 12 〉〉 Download
McL:2 〈〈 c, d | o(c) = 2, o(d) = 3, o(cd) = 22, o(cdcdcdcddcdcddcdd) = 24 〉〉 Download

Presentations

Group Presentation Link
McL 〈 a, b | a2 = b5 = (ab)11 = (ab2)12 = [a, b]5 = [a, b2]6 = (abab−2)7 = [a, b−2ab2ab−1ab(ab2)2abab−1] = [a, b2ab(ab2)2]2 = abab2ab−2abab−1ab2(ab−2ab)2(ab2ab−2ab2)2 = [a, b2ab2ab−1ab2]2 = [a, b2ab]4 = 1 〉 Details
McL:2 〈 c, d | c2 = d3 = (cd)22 = (cdcdcd−1)6 = [c, (dcdcd−1c)2dcd−1cd−1] = [c, d−1cdcd]4 = (cd)6cd−1cd(cdcdcd−1)2(cdcd−1)2cdcdcd−1(cd)4(cd−1)6cdcdcd−1 = (cd)5cd−1(cd)3(cd−1cd−1cd)2cd(cdcd−1cd−1)2(cdcd−1)3cd−1cd(cdcdcd−1)2 = 1 〉 Details

Representations

Representations of McL

Representations of 3.McL

Representations of McL:2

Maximal subgroups

Maximal subgroups of McL

Subgroup Order Index Programs/reps
U4(3) 3 265 920 275 Program: Generators
M22 443 520 2 025 Program: Standard generators
M22 443 520 2 025 Program: Standard generators
U3(5) 126 000 7 128 Program: Standard generators
31+4:2.S5 58 320 15 400 Program: Generators
34:M10 58 320 15 400 Program: Generators
L3(4):21 40 320 22 275 Program: Standard generators
2.A8 40 320 22 275 Program: Generators
24:A7 40 320 22 275 Program: Generators mapping onto standard generators
24:A7 40 320 22 275 Program: Generators mapping onto standard generators
M11 7 920 113 400 Program: Standard generators
51+2:3:8 3 000 299 376 Program: Generators

Maximal subgroups of McL:2

Subgroup Order Index Programs/reps
McL Program: Standard generators
U4(3):2 Program: Generators
U3(5):2 Program: Generators
31+4:4.S5 Program: Generators
34:(M10 × 2) Program: Generators
L3(4):2:2. Program: Generators
2.S8 Program: Generators
Program: Generators
M11 × 2. Program: Generators
Program: Generators
51+2:3:8.2. Program: Generators
22+4:(S3 × S3). Program: Generators

Conjugacy classes

Conjugacy classes of McL

Conjugacy class Centraliser order Power up Class rep(s)
1A 898 128 000 Omitted owing to length.
2A 40 320 4A 6A 6B 8A 10A 12A 14A 14B 30A 30B ababbbabbababbbabbababbbabbababbbabb
3A 29 160 6A 9A 9B 12A 15A 15B 30A 30B abbabbabbabb
3B 972 6B ababbbabbbabbababbbabbbabb
4A 96 8A 12A ababbbabbababbbabb
5A 750 10A 15A 15B 30A 30B (ababbbabbbababbbabbb)3
5B 25 ababbbabbababbbabbbabb
6A 360 12A 30A 30B abbabb
6B 36 ababbbabbbabb
7A 14 7B3 14A 14B ababbababb
7B 14 7A3 14A 14B (ababbababb)3
8A 8 ababbbabb
9A 27 9B2 abababbabababb
9B 27 9A2 abababb
10A 30 30A 30B (ababbbabbb)3
11A 11 11B2 ab
11B 11 11A2 abab
12A 12 abb
14A 14 14B3 ababb
14B 14 14A3 (ababb)3
15A 30 15B7 30A 30B ababbbabbbababbbabbb
15B 30 15A7 30A 30B (ababbbabbbababbbabbb)7
30A 30 30B7 ababbbabbb
30B 30 30A7 (ababbbabbb)7

Download words for class representatives.

Conjugacy classes of McL:2

Conjugacy class Centraliser order Power up Class rep(s)
1A 1 796 256 000
2A 80 640 4A 6A 6B 8A 10A 12A 14A 30A 4B 8B 8C 12B 12C 20A 20B 24A 24B
3A 58 320 6A 9A 12A 15A 30A 12B 24A 24B
3B 1 944 6B 6C 12C
4A 192 8A 12A 8B 8C 24A 24B
5A 1 500 10A 15A 30A 20A 20B
5B 50 10B
6A 720 12A 30A 12B 24A 24B
6B 72 12C
7A 14 14A
8A 16 cdcdcddcdcdcddcdcdcddcdcddcdcdcddcdcdd
9A 27 Omitted owing to length.
10A 60 30A 20A 20B
11A 22 11B2 22A 22B
11B 22 11A2 22A 22B
12A 24 24A 24B
14A 14 cdcdcddcdcdcddcdcdd
15A 30 30A
30A 30 Omitted owing to length.
2B 15 840 6C 10B 22A 22B
4B 1 440 12B 12C 20A 20B
6C 36 cdcdcdd
8B 96 24A 24B
8C 32 Omitted owing to length.
10B 10 cdcdcddcdcdd
12B 36 ccdcdcddcdcdcddcdcdcddcdcddcdcdcddcdcdd
12C 36 Omitted owing to length.
20A 20 20B11 cdcdcddcdcddcdcdd
20B 20 20A11
22A 22 22B7
22B 22 22A7 cd
24A 24 24B13 cdcdcddcdcdcddcdcdcddcdcdd
24B 24 24A13

Download words for class representatives.