# ATLAS: McLaughlin group McL

Order = 898128000 = 27.36.53.7.11.
Mult = 3.
Out = 2.

The following information is available for McL:

### Standard generators

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.

### Black box algorithms

#### 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

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] = [ab2ab(ab2)2]2 = abab2ab-2abab-1ab2(ab-2ab)2(ab2ab-2ab2)2 = [ab2ab2ab-1ab2]2 = [ab2ab]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.

### Representations

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.
• Some 5-modular representations.
• Some 7-modular representations.
• Some 11-modular representations.
• A characteristic 23 representation.
• 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:
• Some irreducibles in characteristic 3:
• Some irreducibles in characteristic 5:
• Some irreducibles in characteristic 7:
• Some irreducibles in characteristic 11:
The representations of 3.McL:2 available are:

### Maximal subgroups

The maximal subgroups of McL are:
The maximal subgroups of McL:2 are:

### Conjugacy classes

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.