# ATLAS: Mathieu group M22

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

The following information is available for M22:

### Standard generators

Standard generators of the Mathieu group M22 are a and b where a has order 2, b is in class 4A, ab has order 11 and ababb has order 11.
There are problems of 'virtue' in defining standard generators for the various covering groups. The ones defined here may change subtly at a later date.
Standard generators of the double cover 2.M22 are preimages A and B where A is in +2A, B is in -4A and AB has order 11 (any two of these conditions imply the third). An equivalent set of conditions is that AB has order 11 and ABABB has order 11.
Standard generators of the triple cover 3.M22 are preimages A and B where A has order 2 and B has order 4.
Standard generators of the fourfold cover 4.M22 are preimages A and B where [A has order 2,] AB has order 11 and ABABB has order 11.
Standard generators of the sixfold cover 6.M22 are preimages A and B where A is in class +2A, and B is in class -4A. (Equivalently, A, B, AB and ABABB have orders 2, 4, 33 and 33 respectively.)
Standard generators of the twelvefold cover 12.M22 are preimages A and B where A has order 2, B has order 4, AB has order 33 and ABABB has order 33.

Standard generators of the automorphism group M22:2 are c and d where c is in class 2B, d is in class 4C and cd has order 11.
Standard generators of the double cover 2.M22.2 are preimages C and D where CD has order 11.
Standard generators of the triple cover 3.M22:2 are preimages C and D where CD has order 11.
Standard generators of the fourfold cover 4.M22.2 are preimages C and D where CD has order 11.
Standard generators of the sixfold cover 6.M22.2 are preimages C and D where CD has order 11.
Standard generators of the twelvefold cover 12.M22.2 are preimages C and D where CD has order 11.

### Black box algorithms

#### Finding generators

To find standard generators for M22:

• Find any element of order 8. Its square is a 4A-element, y say, and its fourth power is a 2A-element, x say.
[The probability of success at each attempt is 1 in 8.]
• Find a conjugate a of x and a conjugate b of y such that ab has order 11 and ababb has order 11.
[The probability of success at each attempt is 64 in 1155 (about 1 in 18).]
This algorithm is available in computer readable format: finder for M22.

To find standard generators for M22.2:

• Find any element of order 12. Its cube is a 4C-element, y say.
[The probability of success at each attempt is 1 in 12.]
• Find any element of order 14. Its seventh power is a 2B-element, x say.
[The probability of success at each attempt is 1 in 7.]
• Find a conjugate c of x and a conjugate d of y such that cd has order 11.
[The probability of success at each attempt is 16 in 55 (about 1 in 3).]
This algorithm is available in computer readable format: finder for M22.2.

#### Checking generators

To check that elements x and y of M22 are standard generators:

• Check o(x) = 2
• Check o(y) = 4
• Check o(xy) = 11
• Check o(xyxyy) = 11
• Check o([x,y]) = 6
This algorithm is available in computer readable format: checker for M22.

To check that elements x and y of M22.2 are standard generators:

• Check o(x) = 2
• Check o(y) = 4
• Check o(xy) = 11
• Check o(xyxyy) = 10
This algorithm is available in computer readable format: checker for M22.2.

### Presentations

Presentations of M22 and M22:2 in terms of their standard generators are given below.

< a, b | a2 = b4 = (ab)11 = (ab2)5 = [a, bab]3 = (ababab-1)5 = 1 >.

< c, d | c2 = d4 = (cd)11 = (cd2)6 = [c, d]4 = (cdcdcd2cd2)3 = 1 >.

These presentations are available in Magma format as follows: M22 on a and b and M22:2 on c and d.

### Representations

Representations are available for the following decorations of M22:

#### M22 and covers

The representations of M22 available are:
• All faithful transitive permutation representations of degree less than 1000 (includes all primitive representations).
• Permutations on 22 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Permutations on 77 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Permutations on 176a points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Permutations on 176b points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Permutations on 231 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Permutations on 330 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Permutations on 462a points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP). - on the cosets of a 2^4:A5 with orbits 6+16.
• Permutations on 462b points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP). - on the cosets of a 2^4:A5 with orbits 1+1+20.
• Permutations on 462c points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP). - on the cosets of a 2^4:A5 with orbits 1+5+16.
• Permutations on 616 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Permutations on 672 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Permutations on 770 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• All faithful irreducibles in characteristic 2.
• All faithful irreducibles in characteristic 3 (up to Frobenius automorphisms).
• All faithful irreducibles in characteristic 5 (up to Frobenius automorphisms and group automorphisms).
• All faithful irreducibles in characteristic 7.
• All faithful irreducibles in characteristic 11.
The representations of 2.M22 available are:
• Permutations on 352 points: A and B (Meataxe), A and B (Meataxe binary), A and B (GAP), A and B (Magma).
• Permutations on 660 points: A and B (Meataxe), A and B (Meataxe binary), A and B (GAP).
• All faithful irreducibles in characteristic 3 (up to field and group automorphisms).
• All faithful irreducibles in characteristic 5 (up to field and group automorphisms).
• All faithful irreducibles in characteristic 7.
• Dimension 10 over GF(7): A and B (Meataxe), A and B (Meataxe binary), A and B (GAP).
• Dimension 56 over GF(7): A and B (Meataxe), A and B (Meataxe binary), A and B (GAP).
• Dimension 120 over GF(7): A and B (Meataxe), A and B (Meataxe binary), A and B (GAP).
• Dimension 126 over GF(49): A and B (Meataxe), A and B (Meataxe binary), A and B (GAP).
• Dimension 126 over GF(49): A and B (Meataxe), A and B (Meataxe binary), A and B (GAP).
• Dimension 154 over GF(49): A and B (Meataxe), A and B (Meataxe binary), A and B (GAP).
• Dimension 154 over GF(49): A and B (Meataxe), A and B (Meataxe binary), A and B (GAP).
• Dimension 210 over GF(7): A and B (Meataxe), A and B (Meataxe binary), A and B (GAP).
• Dimension 252 over GF(7): A and B (Meataxe), A and B (Meataxe binary), A and B (GAP). - reducible over GF(49).
• Dimension 308 over GF(7): A and B (Meataxe), A and B (Meataxe binary), A and B (GAP). - reducible over GF(49).
• Dimension 320 over GF(7): A and B (Meataxe), A and B (Meataxe binary), A and B (GAP).
• All faithful irreducibles in characteristic 11.
• Dimension 10 over Z4: A and B (Magma).
The representations of 3.M22 available are:
• Permutations on 693 points: A and B (Meataxe), A and B (Meataxe binary), A and B (GAP).
• Permutations on 990 points: A and B (Meataxe), A and B (Meataxe binary), A and B (GAP).
• Permutations on 2016 points: A and B (Meataxe), A and B (Meataxe binary), A and B (GAP).
• All faithful irreducibles in characteristic 2 (up to Frobenius automorphisms). [They are all in the z3-cohort.]
• All faithful irreducibles in the z3-cohort in characteristic 5.
• All faithful irreducibles in the z3-cohort in characteristic 7.
• All faithful irreducibles in the z3-cohort in characteristic 11.
The representations of 4.M22 available are:
• Permutations on 4928 points: A and B (Meataxe), A and B (Meataxe binary), A and B (GAP). - on the cosets of an A6 in 4a.L3(4).
• Permutations on 4928 points: A and B (Meataxe), A and B (Meataxe binary), A and B (GAP). - on the cosets of an A6 in 4.M10 (and not in 4.L3(4)).
• All faithful irreducibles in the i-cohort in characteristic 3.
• All faithful irreducibles in the i-cohort in characteristic 5.
• All faithful irreducibles in the i-cohort in characteristic 7.
• All faithful irreducibles in the i-cohort in characteristic 11.
The representations of 6.M22 available are:
• Permutations on 1980 points: A and B (Meataxe), A and B (Meataxe binary), A and B (GAP) - on the cosets of 23:L3(2) (supplied by Sophie Whyte).
• All faithful irreducibles in the z3-cohort in characteristic 5.
• All faithful irreducibles in the z3-cohort in characteristic 7.
• Dimension 54 over GF(7): A and B (Meataxe), A and B (Meataxe binary), A and B (GAP).
• Dimension 66 over GF(7): A and B (Meataxe), A and B (Meataxe binary), A and B (GAP).
• Dimension 126 over GF(49): A and B (Meataxe), A and B (Meataxe binary), A and B (GAP).
• Dimension 126 over GF(49): A and B (Meataxe), A and B (Meataxe binary), A and B (GAP).
• Dimension 210 over GF(7): A and B (Meataxe), A and B (Meataxe binary), A and B (GAP).
• Dimension 210 over GF(49): A and B (Meataxe), A and B (Meataxe binary), A and B (GAP).
• Dimension 210 over GF(49): A and B (Meataxe), A and B (Meataxe binary), A and B (GAP).
• Dimension 252 over GF(7): A and B (Meataxe), A and B (Meataxe binary), A and B (GAP). - reducible over GF(49).
• Dimension 330 over GF(7): A and B (Meataxe), A and B (Meataxe binary), A and B (GAP).
• Dimension 420 over GF(7): A and B (Meataxe), A and B (Meataxe binary), A and B (GAP). - reducible over GF(49).
• All faithful irreducibles in the z3-cohort in characteristic 11.
• Dimension 36 over GF(121): A and B (Meataxe), A and B (Meataxe binary), A and B (GAP).
• Dimension 36 over GF(121): A and B (Meataxe), A and B (Meataxe binary), A and B (GAP). - in the z3**-cohort.
• Dimension 66 over GF(121): A and B (Meataxe), A and B (Meataxe binary), A and B (GAP).
• Dimension 66 over GF(121): A and B (Meataxe), A and B (Meataxe binary), A and B (GAP).
• Dimension 90 over GF(121): A and B (Meataxe), A and B (Meataxe binary), A and B (GAP).
• Dimension 120 over GF(121): A and B (Meataxe), A and B (Meataxe binary), A and B (GAP).
• Dimension 174 over GF(121): A and B (Meataxe), A and B (Meataxe binary), A and B (GAP).
• Dimension 210 over GF(121): A and B (Meataxe), A and B (Meataxe binary), A and B (GAP).
• Dimension 330 over GF(121): A and B (Meataxe), A and B (Meataxe binary), A and B (GAP).
The representations of 12.M22 available are:
• Permutations on 31680a points: A and B (Meataxe), A and B (Meataxe binary), A and B (GAP), A and B (Magma). - on the cosets of an L3(2) in 12a.L3(4).
• All faithful irreducibles in the (i, z3)-cohort in characteristic 5.
• All faithful irreducibles in the (i, z3)-cohort in characteristic 7.
• All faithful irreducibles in the (i, z3)-cohort in characteristic 11.
• All faithful irreducibles in the (i, z3**)-cohort in characteristic 11.

#### M22:2 and covers

The representations of M22:2 available are:
• All faithful pseudoprimitive permutation representations.
• Permutations on 22 points: c and d (Meataxe), c and d (Meataxe binary), c and d (GAP).
• Permutations on 77 points: c and d (Meataxe), c and d (Meataxe binary), c and d (GAP).
• Permutations on 231 points: c and d (Meataxe), c and d (Meataxe binary), c and d (GAP).
• Permutations on 330 points: c and d (Meataxe), c and d (Meataxe binary), c and d (GAP).
• Permutations on 352 points: c and d (Meataxe), c and d (Meataxe binary), c and d (GAP). - pseudoprimitive.
• Permutations on 616 points: c and d (Meataxe), c and d (Meataxe binary), c and d (GAP).
• Permutations on 672 points: c and d (Meataxe), c and d (Meataxe binary), c and d (GAP).
• All faithful irreducibles in characteristic 2.
• All faithful irreducibles in characteristic 3 (up to tensoring with the alternating character).
• All faithful irreducibles in characteristic 5 (up to tensoring with the alternating character).
• All faithful irreducibles in characteristic 7 (up to tensoring with the alternating character).
• All faithful irreducibles in characteristic 11 (up to tensoring with the alternating character).
The representations of 2.M22:2 available are:
The representations of 3.M22:2 available are:
The representations of 4.M22:2 available are:
The representations of 6.M22:2 available are:
The representations of 12.M22:2 available are:
• Dimension 48 over GF(11): C and D (Meataxe), C and D (Meataxe binary), C and D (GAP).
I haven't worked out which one of the two representations of this degree this is yet.

### Maximal subgroups

The maximal subgroups of M22 are as follows. Words provided by Peter Walsh, implemented and checked by Ibrahim Suleiman.
The maximal subgroups of M22:2 are as follows.
The presentation files here and here contain generators for the maximal, and some other subgroups of M22 and M22:2 respectively.

### Conjugacy classes

A set of generators for the maximal cyclic subgroups can be obtained by running this program (W1) or this program (W2) on the standard generators. All conjugacy classes can therefore be obtained as suitable powers of these elements, by running this program (on the output of either the W1- or W2-program).

The word programs give the following class representatives for M22:

• 4B: abababab3 (both W1 and W2).
• 5A: ab2 (both W1 and W2).
• 6A: abab3 (both W1 and W2).
• 7A: ababababab2abab2ab2 (W1 only); abababab3ab2 (W2 only).
• 8A: ababab2ab3 (W1 only); ababab3ab2 (W2 only).
• 11A: ab (both W1 and W2).
• 11B: abab (neither); abab2 (neither); ababab2 (W2 only).
The following are some class representatives of 12.M22:
12.M22-ClassClass representative(s), in terms
of the output of the W1-program
Class representative(s), in terms
of the output of the W2-program
z3.1A
(top central element of order 3)
11A11 = (AB)11 11A11 = (AB)11 or 11B44 = (ABABAB2)44
i.1A
(top central element of order 4)
7A63 11B99 = (ABABAB2)99
[+]4B (class +4B in 3.M22) 4B3 = (ABABABAB3)3 4B3 = (ABABABAB3)3
+5A 5A = AB2 5A = AB2
±6A (class +6A in 6.M22)
+7A 7A36 7A36 or 7A18 = (ABABABAB3AB2)18
+8A 8A21 = (ABABAB2AB3)21 8A9 = (ABABAB3AB2)9
+11A 11A3 = (AB)3 11A3 = (AB)3

The same program (W1) can be used to distinguish central elements in the covers: (DO NOT USE THE W2-PROGRAM FOR THIS.)
The 11th power of 11A is the top central element of order 3, and acts as the scalar z3 in the representations whose character is printed in the Atlas.
The 63rd power of 7A is the top central element of order 4, and acts as the scalar i in the representations whose character is printed in the Atlas.
The 36th power of 7A is in class +7A in all covers and is therefore the element whose character values are printed in the Atlas.
The 21st power of 8A is in class +8A in all covers and is therefore the element whose character values are printed in the Atlas.
The 3rd power of 11A is in class +11A in all covers and is therefore the element whose character values are printed in the Atlas.
Warning: this has not been properly checked, and may be incompatible with information given elsewhere. Also, for 12.M22 mod 11, we should also be giving representations in the cohort where the given central element of order 3 [11A11] acts as the scalar z3** and the given central element of order 4 [7A63] acts as the scalar i.

Go to main ATLAS (version 2.0) page.
Go to sporadic groups page.
Go to old M22 page - ATLAS version 1.
Anonymous ftp access is also available. See here for details.

Version 2.0 created on 23rd January 2001.
Last updated 21.12.04 by SJN.
Information checked to Level 0 on 23.01.01 by RAW.
R.A.Wilson, R.A.Parker and J.N.Bray.