ATLAS: Mathieu group M12
Order = 95040 = 26.33.5.11.
Mult = 2.
Out = 2.
The following information is available for M12:
Standard generators of M12 are a and b where a
is in class 2B, b is in class 3B and ab has order 11.
Standard generators of the double cover 2.M12 are preimages A
and B where A is in class +2B, B has order 6 and AB
has order 11. (Note that any two of these conditions imply the third.)
Standard generators of M12:2 are c and d where
c is in class 2C, d is in class 3A and cd is in class 12A.
(This last condition can be replaced by: cd has order 12 and
cdcdd has order 11.)
Standard generators of either of the double covers 2.M12.2 are
preimages C and D where D has order 3.
A pair of elements automorphic to A, B
can be obtained as
A' = (CDCDCDDCD)3, B' = (CDD)-3(CD)4(CDD)3.
Finding generators
To find standard generators for M12:
- Find any element of order 4 or 8. It powers up to x in
class 2B.
[The probability of success at each attempt is 5/16 (about 1/3).]
- Find any 3-element y, say. If (as is likely) your search finds
a 6-element, z say, before a 3-element, you should take
y = zz.
[The probability that a randomly chosen element has order 3 is 5/108 (about
1/22), with the probability that it then lies in the right class then being
3/5. The probability that an element has order 6 is 1/4, and the probability
that it then squares to the right class is 1/3.]
- Find a conjugate a of x and a conjugate b of y, whose product has order 11.
If ababb has order 6,
then these are standard generators of M12.
Otherwise (when ababb has order 5),
the 3-element is in the wrong conjugacy class, so try again (from the
second step).
[At each attempt, the probability of `success' if y is in the wrong class is 12/55 (about 1/5); if y is in the right class the probability of success is 8/55 (about 1/7).]
An alternative method of finding standard generators for M12:
- Find any element of order 4 or 8. It powers up to x in
class 2B.
[The probability of success at each attempt is 5/16 (about 1/3).]
- Find any element of order 10. It powers up to z in
class 2A.
[The probability of success at each attempt is 1/10.]
- Find a conjugate z' of z such that zz' has order 3
or 6 (so that zz' is in class 3B or 6A). Then zz' powers up to
y in class 3B.
[The probability of success at each attempt is 5/11 (about 1/2).]
- Find a conjugate a of x and a conjugate b of
y, whose product has order 11.
[The probability of success at each attempt is 8/55 (about 1/7).]
- Now a and b are standard generators of M12.
This algorithm is available in computer readable format:
finder for M12.
To find standard generators for M12.2:
- Find any element of order 8. It powers up to z in
class 2B.
[The probability of success at each attempt is 1/8.]
- Find any element w of order 12.
[The probability of success at each attempt is 1/4.]
- Find conjugates z' of z and w' of w such that
z'w' has order 10. Then x = (z'w')^5 is a
2C-element.
[The probability of success at each attempt is 32/165 (about 1/5).]
- Find conjugates z' of z and x' of x such that
z'x' has order 12 (necessarily in classes 12B or 12C). Then
y = (z'x')^4 is a 3A-element.
[The probability of success at each attempt is 16/33 (about 1/2).]
- Find conjugates c of x and d of y such that
cd has order 12 and cdcdd has order 11.
[The probability of success at each attempt is 3/22 (about 1/7).]
- Now c and d are standard generators of M12:2.
This algorithm is available in computer readable format:
finder for M12.2.
Checking generators
To check that elements x and y of M12
are standard generators:
- Check o(x) = 2
- Check o(y) = 3
- Check o(xy) = 11
- Check o(xyxyxyy) = 6
This algorithm is available in computer readable format:
checker for M12.
To check that elements x and y of M12.2
are standard generators:
- Check o(x) = 2
- Check o(y) = 3
- Check o(xy) = 12
- Check o(xyxyxyxyy) = 6
This algorithm is available in computer readable format:
checker for M12.2.
Presentations of M12 and M12:2 in terms of their standard generators are given below.
< a, b | a2 = b3 =
(ab)11 = [a, b]6 =
(ababab-1)6 = 1 >.
< c, d | c2 = d3 =
(cd)12 =
(cd)5[c, d](cd-1)3cd[c, d-1]2cdcd(cd-1)3[c, d-1] = 1 >.
These presentations are available in Magma format as follows:
M12 on a and b,
2M12 on A and B,
2M12 on A'' and B'',
M12:2 on c and d [v1] and
M12:2 on c and d [v2].
The representations of M12 available are:
- Permutation representations.
-
Permutations on 12a points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Permutations on 12b points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Permutations on 66a points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Permutations on 66b 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).
-
Dimension 16 over GF(4):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 16 over GF(4):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 44 over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 144 over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
- All faithful irreducibles in characteristic 3 (up to automorphisms).
-
Dimension 10 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 15 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 34 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 45 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 45 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 54 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).
- All faithful irreducibles in characteristic 5 (up to automorphisms).
-
Dimension 11 over GF(5):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 16 over GF(5):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 45 over GF(5):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 55a over GF(5):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 55c over GF(5):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 66 over GF(5):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 78 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 120 over GF(5):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
- All faithful irreducibles in characteristic 11 (up to automorphisms).
-
Dimension 11 over GF(11):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 16 over GF(11):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 29 over GF(11):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 53 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 55 over GF(11):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 66 over GF(11):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 91 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 176 over GF(11):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
The representations of 2.M12 available are:
-
Permutations on 24a points:
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
-
Dimension 6b over GF(3):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
-
Dimension 12 over GF(5):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
The representations of M12:2 available are:
-
Permutations on 24 points:
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
- All 2-modular irreducibles.
-
Dimension 10 over GF(2):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
-
Dimension 32 over GF(2):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
-
Dimension 44 over GF(2):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
-
Dimension 144 over GF(2):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
- All 3-modular irreducibles (up to sign).
-
Dimension 20 over GF(3):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
-
Dimension 30 over GF(3):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
-
Dimension 34 over GF(3):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
-
Dimension 45 over GF(3):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
-
Dimension 54 over GF(9):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
-
Dimension 90 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).
- All 5-modular irreducibles (up to sign).
-
Dimension 22 over GF(5):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
-
Dimension 32 over GF(5):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
-
Dimension 45 over GF(5):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
-
Dimension 54 over GF(5):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
-
Dimension 66 over GF(5):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
-
Dimension 78 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 110 over GF(5):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
-
Dimension 120 over GF(25):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
- All 11-modular irreducibles (up to sign).
-
Dimension 16 over GF(11):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
-
Dimension 22 over GF(11):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
-
Dimension 29 over GF(11):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
-
Dimension 53 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 66 over GF(11):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
-
Dimension 91 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 110 over GF(11):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
-
Dimension 176 over GF(11):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
The representations of 2.M12:2 available are:
-
Permutations on 48 points:
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).
- All irreducibles in characteristic 3 (up to sign):
-
Dimension 10 over GF(3):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).
-
Dimension 10 over GF(3):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).
-
Dimension 10 over GF(3):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).
-
Dimension 12 over GF(3):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).
-
Dimension 88 over GF(3):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).
-
Dimension 168 over GF(3):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).
- All irreducibles in characteristic 5 (up to sign):
-
Dimension 10 over GF(25):
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 12 over GF(25):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).
-
Dimension 32 over GF(25):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).
-
Dimension 120 over GF(5):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).
-
Dimension 220 over GF(5):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).
-
Dimension 320 over GF(5):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).
- All irreducibles in characteristic 11 (up to sign):
-
Dimension 10 over GF(11):
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).
-
Dimension 12 over GF(11):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).
-
Dimension 32 over GF(11):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).
-
Dimension 88 over GF(11):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).
-
Dimension 108 over GF(11):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).
-
Dimension 220 over GF(11):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).
The maximal subgroups of M12 are:
-
M11, with standard generators
(ab)^-2a(ab)^2,
(abb)^-2(ababababbabbababb)^2(abb)^2.
-
M11, with standard generators
a,
(abb)^-3(ababababbababbabb)^2(abb)^3.
-
A6.2.2, with generators
(bab)^-1abab,
(abb)^-1ababababbabbababbabb
.
-
A6.2.2, with generators
(ab)^-3a(ab)^3,
(abb)^-2ab(ababb)^2abb(abb)^2.
-
L2(11), with standard generators
(abbabab)^3,
(abb)^-2b(abb)^2.
-
3^2:2S4, with generators
here.
-
3^2:2S4, with generators
here.
-
2 × S5, with generators (mapping to standard
generators of S5)
(ab)^-1(abababb)^3ab,
(abb)^-3(abababbababbabb)(abb)^3.
Note that a pair of generators of 2 × S5 mapping
onto standard generators of S5 is essentially unique; thus such a
pair may be regarded as being standard generators for 2 × S5.
-
2^1+4:S3, with generators
here, or
here.
- 4^2:D12, with generators
here.
- A4 × S3, with generators
(ab)^-1(abababb)^3ab,
(abb)^-1babb.
The maximal subgroups of M12:2 are:
A set of generators for the maximal cyclic subgroups of M12 can be obtained
by running this program on the standard
generators. All conjugacy classes can therefore be obtained as suitable
powers of these elements.
There are no problems of algebraic conjugacy.
Representatives of the 15 conjugacy classes of M12 are also given below. The first choice of class representative for classes 6A, 6B, 8A, 8B,
10A and 11A is that which is produced by the above word program.
- 1A: identity [or a2].
- 2A: (ababab2)3.
- 2B: a.
- 3A: [a, b]2.
- 3B: b.
- 4A: ababab2abab2ab2.
- 4B: ababab2ab2abab2.
- 5A: [a, bab].
- 6A: ababab2.
- 6B: abab2 or [a, b].
- 8A: abababab2abab2ab2.
- 8B: abababab2ab2abab2.
- 10A: abababab2ab2.
- 11A: ab.
- 11B: ab2.
The preimages of classes 2A, 4A, 4B and 6A do not split in 2.M12, and the
resulting element orders are 4, 4, 4 and 12. We give representatives of the other 11 classes up to multiplication by the central involution. All of this
is forced by the 6-dimensional 3-modular representations and the 24-point permutation representations.
- +1A [order 1]: identity [or A2].
- +2B [order 2]: A.
- +3A [order 3]: [A, B]2.
- -3B [order 6]: B.
- +5A [order 5]: [A, BAB].
- -6B [order 6]: ABAB2; [A, B] is in class +6B [order 6].
- +8A [order 8]: ABABABAB2ABAB2AB2.
- -8B [order 8]: ABABABAB2AB2ABAB2.
- -10A [order 20]: ABABABAB2AB2.
- +11A [order 11]: AB.
- -11B [order 22]: AB2; AB-1 is in class +11B [order 11].
A set of generators for the maximal cyclic subgroups of M12.2 can be obtained
by running this program on the standard
generators. All conjugacy classes can therefore be obtained as suitable
powers of these elements.
The notation has been chosen so that the words for the representatives of classes 12A, 12C and 8AB
give elements of 2M12.2 in classes +12A, +12C and +8AB.
Go to main ATLAS (version 2.0) page.
Go to sporadic groups page.
Go to old M12 page - ATLAS version 1.
Anonymous ftp access is also available.
See here for details.
Version 2.0 created on 23rd January 2001.
Last updated 10.1.05 by SJN.
Information checked to
Level 0 on 23.01.01 by RAW.
R.A.Wilson, R.A.Parker and J.N.Bray.