ATLAS: Mathieu group M_{22}
Order = 443520 = 2^{7}.3^{2}.5.7.11.
Mult = 12.
Out = 2.
The following information is available for M_{22}:
Standard generators of the Mathieu group M_{22} 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.M_{22} 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.M_{22} are preimages A and
B where A has order 2 and B has order 4.
Standard generators of the fourfold cover 4.M_{22} 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.M_{22} 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.M_{22} 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 M_{22}: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.M_{22}.2 are preimages
C and D where CD
has order 11.
Standard generators of the triple cover 3.M_{22}:2 are preimages
C and D where CD
has order 11.
Standard generators of the fourfold cover 4.M_{22}.2 are preimages
C and D where CD
has order 11.
Standard generators of the sixfold cover 6.M_{22}.2 are preimages
C and D where CD
has order 11.
Standard generators of the twelvefold cover 12.M_{22}.2 are preimages
C and D where CD
has order 11.
Finding generators
To find standard generators for M_{22}:
 Find any element of order 8. Its square is a 4Aelement, y say,
and its fourth power is a 2Aelement, 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 M_{22}.
To find standard generators for M_{22}.2:
 Find any element of order 12. Its cube is a 4Celement, y say.
[The probability of success at each attempt is 1 in 12.]
 Find any element of order 14. Its seventh power is a 2Belement, 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 M_{22}.2.
Checking generators
To check that elements x and y of M_{22}
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 M_{22}.
To check that elements x and y of M_{22}.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 M_{22}.2.
Presentations of M_{22} and M_{22}:2 in terms of their
standard generators are given below.
< a, b  a^{2} = b^{4} =
(ab)^{11} = (ab^{2})^{5} =
[a, bab]^{3} =
(ababab^{1})^{5} = 1 >.
< c, d  c^{2} = d^{4} =
(cd)^{11} = (cd^{2})^{6} =
[c, d]^{4} =
(cdcdcd^{2}cd^{2})^{3} = 1 >.
These presentations are available in Magma format as follows:
M_{22} on a and b and
M_{22}:2 on c and d.
Representations are available for the following decorations of M_{22}:
M_{22} and covers
The representations of M_{22} 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.

Dimension 10 over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP),
a and b (Magma).
 the `code' module.

Dimension 10 over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP),
a and b (Magma).
 the `cocode' module.

Dimension 34 over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 70 over GF(4):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 70 over GF(4):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 98 over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 All faithful irreducibles in characteristic 3 (up to Frobenius automorphisms).

Dimension 21 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 45 over GF(9):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 49 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 49 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 55 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 99 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 210 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 231 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 All faithful irreducibles in characteristic 5 (up to Frobenius automorphisms
and group automorphisms).

Dimension 21 over GF(5):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 45 over GF(25):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 55 over GF(5):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 98 over GF(5):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 133 over GF(5):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 210 over GF(5):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 280 over GF(5):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 385 over GF(5):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 All faithful irreducibles in characteristic 7.

Dimension 21 over GF(7):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 45 over GF(7):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 54 over GF(7):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 154 over GF(7):
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 231 over GF(7):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 280 over GF(49):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 280 over GF(49):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 385 over GF(7):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 All faithful irreducibles in characteristic 11.

Dimension 20 over GF(11):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 45 over GF(11):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 45 over GF(11):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 55 over GF(11):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 99 over GF(11):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 154 over GF(11):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 190 over GF(11):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 231 over GF(11):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 385 over GF(11):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
The representations of 2.M_{22} 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).

Dimension 10 over GF(9):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 10 over GF(9):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 56 over GF(3):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 120 over GF(3):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 126 over GF(3):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 154 over GF(9):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 210 over GF(3):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
 All faithful irreducibles in characteristic 5 (up to field and group automorphisms).

Dimension 10 over GF(25):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 10 over GF(25):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 28 over GF(5):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 28 over GF(5):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 120 over GF(5):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 126 over GF(5):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 210 over GF(5):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 330 over GF(5):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 440 over GF(5):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
 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 GF(11):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 10 over GF(11):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 56 over GF(11):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 64 over GF(11):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 126 over GF(11):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 154 over GF(121):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 154 over GF(121):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 308 over GF(11):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
 reducible over GF(121).

Dimension 330 over GF(11):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 440 over GF(11):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 10 over Z_{4}:
A and B (Magma).
The representations of 3.M_{22} 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 z3cohort.]

Dimension 6 over GF(4):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 15 over GF(4):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 45 over GF(4):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 45 over GF(4):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 84 over GF(4):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 384 over GF(4):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
 All faithful irreducibles in the z3cohort in characteristic 5.

Dimension 21 over GF(25):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 45 over GF(25):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 45 over GF(25):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 78 over GF(25):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 105 over GF(25):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 105 over GF(25):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 153 over GF(25):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 210 over GF(25):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 231 over GF(25):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 330 over GF(25):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
 All faithful irreducibles in the z3cohort in characteristic 7.

Dimension 21 over GF(7):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 45 over GF(7):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 99 over GF(7):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 105 over GF(49):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 105 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(7):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
 reducible over GF(49).

Dimension 231 over GF(7):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 231 over GF(7):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 285 over GF(7):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
 All faithful irreducibles in the z3cohort in characteristic 11.

Dimension 21 over GF(121):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 21 over GF(121):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
 in the z3**cohort.

Dimension 45 over GF(121):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 45 over GF(121):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 84 over GF(121):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 99 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 231 over GF(121):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 231 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 4.M_{22} 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 icohort in characteristic 3.

Dimension 56 over GF(9):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 56 over GF(9):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 64 over GF(9):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 144 over GF(9):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 144 over GF(9):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 160 over GF(9):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 160 over GF(9):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
 All faithful irreducibles in the icohort in characteristic 5.

Dimension 56 over GF(25):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 56 over GF(25):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 88 over GF(25):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 88 over GF(25):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 160 over GF(5):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 160 over GF(5):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 560 over GF(5):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
 All faithful irreducibles in the icohort in characteristic 7.

Dimension 16 over GF(49):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 56 over GF(49):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 56 over GF(49):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 144 over GF(49):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 160 over GF(49):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 560 over GF(49):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
 All faithful irreducibles in the icohort in characteristic 11.

Dimension 56 over GF(121):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 56 over GF(121):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 144 over GF(121):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 144 over GF(121):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 160 over GF(121):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 176 over GF(121):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
The representations of 6.M_{22} available are:
 Permutations on 1980 points:
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP)
 on the cosets of 2^{3}:L_{3}(2) (supplied by Sophie Whyte).
 All faithful irreducibles in the z3cohort in characteristic 5.

Dimension 66 over GF(25):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 66 over GF(25):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 120 over GF(25):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 126 over GF(25):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 126 over GF(25):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 210 over GF(25):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 210 over GF(25):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 210 over GF(25):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 330 over GF(25):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
 All faithful irreducibles in the z3cohort 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 z3cohort 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.M_{22} 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.

Dimension 48 over GF(25):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 120 over GF(25):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 120 over GF(25):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 144 over GF(25):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 144 over GF(25):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 336 over GF(25):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
 All faithful irreducibles in the (i, z3)cohort in characteristic 7.

Dimension 120 over GF(49):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 120 over GF(49):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 144 over GF(49):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 336 over GF(49):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 336 over GF(49):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
 All faithful irreducibles in the (i, z3)cohort in characteristic 11.

Dimension 24 over GF(121):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP),
A and B (Magma).

Dimension 24 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 120 over GF(121):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 336 over GF(121):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
 All faithful irreducibles in the (i, z3**)cohort in characteristic 11.

Dimension 96 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 120 over GF(121):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 144 over GF(121):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 144 over GF(121):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
M_{22}:2 and covers
The representations of M_{22}: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.

Dimension 10 over GF(2):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP),
c and d (Magma).
 the `code' module.

Dimension 10 over GF(2):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP),
c and d (Magma).
 the `cocode' module.

Dimension 34 over GF(2):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 98 over GF(2):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 140 over GF(2):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
 All faithful irreducibles in characteristic 3 (up to tensoring with the alternating character).

Dimension 21 over GF(3):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 45 over GF(9):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 45 over GF(9):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 98 over GF(3):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 55 over GF(3):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 99 over GF(3):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 210 over GF(3):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 231 over GF(3):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
 All faithful irreducibles in characteristic 5 (up to tensoring with the alternating character).

Dimension 21 over GF(5):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 45 over GF(25):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 45 over GF(25):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 55 over GF(5):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 98 over GF(5):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 133 over GF(5):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 210 over GF(5):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 385 over GF(5):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 560 over GF(5):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
 All faithful irreducibles in characteristic 7 (up to tensoring with the alternating character).

Dimension 21 over GF(7):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 45 over GF(7):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 54 over GF(7):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 154 over GF(7):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 210 over GF(7):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 231 over GF(7):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 385 over GF(7):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 560 over GF(7):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
 All faithful irreducibles in characteristic 11 (up to tensoring with the alternating character).

Dimension 20 over GF(11):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 45 over GF(11):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 45 over GF(11):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 55 over GF(11):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 99 over GF(11):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 154 over GF(11):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 190 over GF(11):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 231 over GF(11):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 385 over GF(11):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
The representations of 2.M_{22}:2 available are:

Dimension 10 over GF(9):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).

Dimension 10 over GF(25):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).

Dimension 10 over GF(7):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).

Dimension 10 over GF(11):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).
 C and D as
20 x 20 integral matrices, in GAP format (kindly provided by N.J.A.Sloane and G.Nebe
from their library of lattices).
The representations of 3.M_{22}:2 available are:

Dimension 12 over GF(2):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).
The representations of 4.M_{22}:2 available are:

Dimension 32 over GF(7):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).
The representations of 6.M_{22}:2 available are:

Dimension 72 over GF(11):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).
The representations of 12.M_{22}: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.
The maximal subgroups of M_{22} are as follows. Words provided by Peter Walsh,
implemented and checked by Ibrahim Suleiman.

L_{3}(4), with generators
a, b^{abababababbababb}.

2^{4}:A_{6}, with generators
a, (ababb)^{6}(ab)^{4}abb(ab)^{4}(ababb)^{6}.

A_{7}, with generators
a, (ababb)^{7}(ab)^{2}abb(ab)^{2}(ababb)^{7}.

A_{7}, with generators
(abb)^{abababababb}.

2^{4}:S_{5}, with generators
a, abab^{2}abab(abab^{2})^{2}ab^{2}ab.

2^{3}:L_{3}(2), with generators
a, (ababb)^{3}(ab)^{2}b(ab)^{2}(ababb)^{3}.

M_{10} = A_{6}.2, with generators
a, (abab^{2}ab^{3}ab)^{b} or
here.

L_{2}(11), with generators
a, (ababb)^{3}(ab)^{2}abb(ab)^{2}(ababb)^{3}.
The maximal subgroups of M_{22}:2 are as follows.

M_{22}, with standard generators
d^{2}, (cdcddcdcd)^{2} [new version] or
d^{2}, d^{1}(cdcddcdcd)^{2}d [old version].

L_{3}(4):2_{2},
with standard generators c, ((cddcd)^{2})^{ddcdcd},
and generators c, cdcd^{3}cd^{2}cd.

2^{4}:S_{6},
with generators c, ((cddcd)^{2})^{ddcd}.

2^{5}:S_{5},
with generators c, (dcddcdcd)^{cd}.

2^{3}:L_{3}(2) × 2,
with generators d^{2}, (cdd)^(dcdcd^{3}cd^{3}cd^{2}cd^{3}cdc),
and generators c, d^{cddcd}.

Aut(A_{6}) = A_{6}.2^{2},
with standard generators c, cddcddcd.

L_{2}(11):2, with standard generators
(cddcd)^{5}, (cddcdd)^{dcdcdd},
and generators cd^{2}cd, d.
The presentation files here and here contain generators for the maximal, and some other subgroups of M_{22} and M_{22}:2 respectively.
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 W2program).
The word programs give the following class representatives for M_{22}:
 4B: abababab^{3} (both W1 and W2).
 5A: ab^{2} (both W1 and W2).
 6A: abab^{3} (both W1 and W2).
 7A: ababababab^{2}abab^{2}ab^{2} (W1 only); abababab^{3}ab^{2} (W2 only).
 8A: ababab^{2}ab^{3} (W1 only); ababab^{3}ab^{2} (W2 only).
 11A: ab (both W1 and W2).^{ }
 11B: abab (neither); abab^{2} (neither); ababab^{2} (W2 only).
The following are some class representatives of 12.M_{22}:
12.M_{22}Class  Class representative(s), in terms of the output of the W1program  Class representative(s), in terms of the output of the W2program 
z3.1A (top central element of order 3) 
11A^{11} = (AB)^{11} 
11A^{11} = (AB)^{11} or 11B^{44} = (ABABAB^{2})^{44} 
i.1A (top central element of order 4) 
7A^{63}

11B^{99} = (ABABAB^{2})^{99} 
[+]4B (class +4B in 3.M22) 
4B^{3} = (ABABABAB^{3})^{3} 
4B^{3} = (ABABABAB^{3})^{3} 
+5A 
5A = AB^{2} 
5A = AB^{2} 
±6A (class +6A in 6.M22) 


+7A 
7A^{36} 
7A^{36} or 7A^{18} = (ABABABAB^{3}AB^{2})^{18} 
+8A 
8A^{21} = (ABABAB^{2}AB^{3})^{21} 
8A^{9} = (ABABAB^{3}AB^{2})^{9} 
+11A 
11A^{3} = (AB)^{3} 
11A^{3} = (AB)^{3} 
The same program (W1) can be used to distinguish central elements in the covers:
(DO NOT USE THE W2PROGRAM 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 [11A^{11}] acts as the scalar z3^{**} and the
given central element of order 4 [7A^{63}] 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.