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:

This algorithm is available in computer readable format: finder for McL.

To find standard generators for McL.2:

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:

This algorithm is available in computer readable format: checker for McL.

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

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: The representations of 3.McL available are: The representations of McL:2 available are: 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.


Main ATLAS page Go to main ATLAS (version 2.0) page.
Sporadic groups page Go to sporadic groups page.
Old McL page Go to old McL page - ATLAS version 1.
ftp access 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.