ATLAS: McLaughlin group McL
Order = 898128000 = 27.36.53.7.11.
Mult = 3.
Out = 2.
The following information is available for McL:
Standard generators of the McLaughlin group McL are a and b
where a is in class 2A, b is in class 5A, ab has order 11
and ababababbababbabb has order 7.
Standard generators of the triple cover 3.McL are preimages A and
B where A has order 2 and B has order 5.
The outer automorphism is achieved by this program.
Standard generators of the automorphism group McL:2 are c and d
where c is in class 2B, d is in class 3B, cd has order 22 and cdcdcdcddcdcddcdd has order 24.
Standard generators of 3.McL:2 are preimages C and D where
CDCDCDDCD has order 11.
A pair of generators conjugate to
a, b can be obtained as
a' = (cd)^{-1}(cdcdcddcdcdcddcd)^{12}cd,
b' = (cdd)^{-3}(cdcdd)^{3}(cdd)^3.
Finding generators
To find standard generators for McL:
- Find any element x of order 2.
- Find any element of order 10, 15 or 30. This powers up to a 5A-element, y say.
- Find a conjugate a of x and a conjugate b of
y, whose product has order 11,
such that (ab)2(ababb)2abb has order 7.
This algorithm is available in computer readable format:
finder for McL.
To find standard generators for McL.2:
- Find any element of order 22. It powers up to a 2B-element,
x say.
- Classes 6A, 6B, 6C occur in the ratio 1:10:20; thus 30/31 of elements of
order 6 square into class 3B. All outer elements of order 6 square
into class 3B, and 1/18 of outer elements have order 6; thus if you can
restrict your search to outer elements you can find a 3B-element. When you find
a 3B-element, call it y, say.
- Find a conjugate c of x and a conjugate d of
y, whose product has order 22,
such that (cd)2(cdcdd)2cdd has order 24.
[NB: You will not be able to find conjugates c of x and
d of y whose product has order 22 if y is in class 3A.]
This algorithm is available in computer readable format:
finder for McL.2.
Checking generators
To check that elements x and y of McL
are standard generators:
- Check o(x) = 2
- Check o(y) = 5
- Check o(xy) = 11
- Check o(xyxyxyxyyxyxyyxyy) = 7
- Check o(xyy) = 12
This algorithm is available in computer readable format:
checker for McL.
To check that elements x and y of McL.2
are standard generators:
- Check o(x) = 2
- Check o(y) = 3
- Check o(xy) = 22
- Check o(xyxyxyxyyxyxyyxyy) = 24
This algorithm is available in computer readable format:
checker for McL.2.
Presentations of McL and McL:2 in terms of their standard
generators are given below.
< 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 >.
< 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 >.
These presentations are available in Magma format as follows:
McL on a and b and
McL:2 on c and d.
The representations of McL available are:
- All primitive permutation representations.
-
Permutations on 275 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Permutations on 2025a points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Permutations on 2025b points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Permutations on 7128 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Permutations on 15400a points - the cosets of 31+42S5:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Permutations on 15400b points - the cosets of 34M10:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Permutations on 22275a points - the cosets of L3(4).2:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Permutations on 22275b points - the cosets of 2A8:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Permutations on 22275c points - the cosets of 24:A7:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Permutations on 22275d points - the cosets of the other 24:A7:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Permutations on 113400 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Permutations on 299376 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
- Some 2-modular representations.
-
Dimension 22 over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 230 over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 748 over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 748 over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
The above two representations have been swapped relative to Version 1.
-
Dimension 896 over GF(4):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 896 over GF(4):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
- Some 3-modular representations.
-
Dimension 21 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 104 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 104 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 560 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 605 over GF(9):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 605 over GF(9):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
- Some 5-modular representations.
-
Dimension 21 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 230 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).
-
Dimension 896 over GF(5):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 1200 over GF(25):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
- Some 7-modular representations.
-
Dimension 22 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 252 over GF(7):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 770a over GF(49):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 770b over GF(49):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 896a over GF(49):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 896b over GF(49):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
- Some 11-modular representations.
-
Dimension 22 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 251 over GF(11):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 770a over GF(121):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 770b over GF(121):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 896 over GF(11):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
- A characteristic 23 representation.
-
Dimension 896b over GF(23):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
- A characteristic 0 representation.
-
Dimension 22 over Z:
a and b (Magma).
The representations of 3.McL available are:
-
Permutations on 66825 points - the cosets of 2.A8:
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
-
Permutations on 103950 points - the cosets of a U4(2):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
-
Permutations on 340200 points:
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
-
Dimension 126 over GF(4):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
-
Dimension 396 over GF(4):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
-
Dimension 42 over GF(3) - uniserial 21.21:
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 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 153 over GF(25):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
-
Dimension 639 over GF(25):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
-
Dimension 846 over GF(25):
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 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 126 over GF(121):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
The representations of McL:2 available are:
-
Permutations on 275 points:
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
-
Permutations on 4050 points:
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
-
Permutations on 7128 points:
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
-
Permutations on 22275 points - the cosets of L3(4).22:
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
-
Permutations on 44550 points:
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
- Some irreducibles in characteristic 2:
-
Dimension 22 over GF(2):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
-
Dimension 230 over GF(2):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
-
Dimension 896 over GF(4):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
-
Dimension 896 over GF(4):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
-
Dimension 1496 over GF(2):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
- Some irreducibles in characteristic 3:
-
Dimension 21 over GF(3):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
-
Dimension 104 over GF(3):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
-
Dimension 104 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 560 over GF(3):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
- Some irreducibles in characteristic 5:
-
Dimension 21 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 230 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).
-
Dimension 896 over GF(5):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
-
Dimension 896 over GF(5):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
- Some irreducibles in characteristic 7:
-
Dimension 22 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 252 over GF(7):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
-
Dimension 896 over GF(49):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
- Some irreducibles in characteristic 11:
-
Dimension 22 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 251 over GF(11):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
-
Dimension 896 over GF(11):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
The representations of 3.McL:2 available are:
-
Dimension 252 over GF(4):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).
-
Dimension 90 over GF(5):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).
-
Dimension 306 over GF(5):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).
-
Dimension 1278 over GF(5):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).
The maximal subgroups of McL are:
-
U4(3), with generators
a,
(abb)^-5(bababababbababb)(abb)^5.
-
M22, with standard generators
(abb)^-2bbabb,
(ab)^4(abababbabababbab)(ab)^7.
-
M22, with standard generators
ababa(ab)^-2,
(abb)^4(abababbabababbab)(abb)^8.
-
U3(5), with standard generators
(ababababbababb)^-3(ababababbababbabbabb
abababababbababbabbabbababb)^2(ababababbababb)^3,
(abbababababbababbabb)^15b(abbababababbababbabb)^15.
-
3^1+4:2.S5, with generators
(ab)^3(abb)^2(ab)^-3,
(abb)^-4ababbbabb(abb)^4.
-
3^4:M10, with generators
(abb)^5a(abb)^-5,
(ababb)^-3(babababbababb)(ababb)^3.
-
L3(4):2, with standard generators
(abb)^2a(abb)^-2,
(ababb)^-5(bababababbababb)(ababb)^5.
-
2.A8, with generators
ababb,
((ababb)^7(abbab)^7)^-2(abbab)((ababb)^7(abbab)^7)^2.
-
2^4:A7, with generators
here, mapping to standard generators of A7.
(Words corrected on 06.01.03: old words generated 2.A7.)
-
2^4:A7, with generators
here, mapping to standard generators of A7.
(Words corrected on 06.01.03: old words generated 2.A7.)
-
M11, with standard generators
b^-1ab,
(abb)^8(abababbabababbab)(abb)^4.
-
5^1+2:3:8, with generators
(ababb)^-4(abb)^4(ababb)^4,
(abbab)^-4ababbbabb(abbab)^4.
The maximal subgroups of McL:2 are:
A set of generators for the maximal cyclic subgroups of McL 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 top central element of order 3 in 3.McL is the 11th power of the
element called 11A.
A set of generators for the maximal cyclic subgroups of McL.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.
Note that the definitions of some classes have been changed (24/11/04) for compatibility
with McL: in particular cd is in class 22B not 22A.
This affects the labelling of a few matrix representations.
Go to main ATLAS (version 2.0) page.
Go to sporadic groups page.
Go to old McL page - ATLAS version 1.
Anonymous ftp access is also available.
See here for details.
Version 2.0 created on 21st June 2000.
Last updated 21.12.04 by SJN.
Information checked to
Level 0 on 21.06.00 by RAW.
R.A.Wilson, R.A.Parker and J.N.Bray.