Order = 443520 = 27.32.5.7.11.
Mult = 12.
Out = 2.

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Porting notes

Porting incomplete.

Standard generators

Standard generators of M22 are a, b where a has order 2, b is in class 4A, ab has order 11 and ababb has order 11.

Standard generators of 2.M22 are preimages A, B where A is in class +2A and B is in class -4A. Alternatively: A is in class +2A and AB has order 11 or B is in class -4A and AB has order 11 or AB has order 11 and ABABB has order 11.

Standard generators of 3.M22 are preimages A, B where A has order 2 and B has order 4.

Standard generators of 4.M22 are preimages A, B where A has order 2, AB has order 11 and ABABB has order 11.

Standard generators of 6.M22 are preimages A, B where A is in class +2A and B is in class -4A. Alternatively: A has order 2, B has order 4, AB has order 33 and ABABB has order 33.

Standard generators of 12.M22 are preimages A, B where A has order 2, B has order 4, AB has order 33 and ABABB has order 33.

Black box algorithms

Finding generators

Group Algorithm File
M22
  1. Find any element of order 8. Its square is a 4A-element, y say, and its fourth power is a 2A-element, x say
    [Probability of success on each attempt is 1/8]
  2. Find a conjugate a of x and a conjugate b of y such that ab has order 11 and ababb has order 11.
    [Probability of success on each attempt is 64/1155 (approx. 1/18)]
 Download

Checking generators (semi-presentations)

Group Semi-presentation File
M22 〈〈 a, b | o(a) = 2, o(b) = 4, o(ab) = 11, o(ababb) = 11, o([a,b]) = 6 〉〉 Download

Presentations

Group Presentation Link
M22 〈 a, b | a2 = b4 = (ab)11 = (ab2)5 = [a,bab]3 = (ababab−1)5 = 1 〉 Details

Representations

Representations of M22

Representations of 2.M22

Representations of 3.M22

Representations of 4.M22

Representations of 6.M22

Representations of 12.M22

Maximal subgroups

Maximal subgroups of M22

Subgroup Order Index Programs/reps
L3(4) 20 160 22 Program: Generators
24:A6 5 760 77 Program: Generators
A7 2 520 176 Program: Generators
A7 2 520 176 Program: Generators
24:S5 1 920 231 Program: Generators
23:L3(2) 1 344 330 Program: Generators
M10 = A6.23 720 616 Program: Generators
Program: Generators
L2(11) 660 672 Program: Generators

Conjugacy classes

Conjugacy classes of M22

Conjugacy class Centraliser order Power up Class rep(s)
1A 443 520 ababababbbababababbbababababbbababababbb
2A 384 4A 4B 6A 8A ababababbbababababbb
3A 36 6A ababbbababbb
4A 32 8A abababbbabbabababbbabb
4B 16 ababababbb
5A 5 abb
6A 12 ababbb
7A 7 7B3 ababababbbabb
7B 7 7A3 (ababababbbabb)3
8A 8 abababbbabb
11A 11 11B2 ab
11B 11 11A2 abab

Download words for class representatives.