# ATLAS: McLaughlin group McL

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

### 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 pre-images 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 pre-images 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

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.
To find standard generators for McL.2:
• Find any element of order 22. It powers up to a 2B-element.
• I don't know how to find a 3B-element.
• Find a conjugate a of x and a conjugate b of y, whose product has order 22, such that (ab)^2(ababb)^2abb has order 24.

### Representations

The representations of McL available are
• Some 2-modular representations
• a and b as 22 × 22 matrices over GF(2).
• a and b as 230 × 230 matrices over GF(2).
• a and b as 748 × 748 matrices over GF(2).
• a and b as 748 × 748 matrices over GF(2) - the dual of the above.
• Some 3-modular representations
• a and b as 21 × 21 matrices over GF(3).
• a and b as 104 × 104 matrices over GF(3).
• a and b as 104 × 104 matrices over GF(3) - the dual of the above.
• a and b as 210 × 210 matrices over GF(3).
• a and b as 560 × 560 matrices over GF(3).
• Some 5-modular representations
• a and b as 21 × 21 matrices over GF(5).
• a and b as 210 × 210 matrices over GF(5).
• a and b as 230 × 230 matrices over GF(5).
• a and b as 560 × 560 matrices over GF(5).
• a and b as 1200 × 1200 matrices over GF(25).
• Some 7-modular representations
• a and b as 22 × 22 matrices over GF(7).
• a and b as 231 × 231 matrices over GF(7).
• a and b as 252 × 252 matrices over GF(7).
• Some 11-modular representations
• a and b as 22 × 22 matrices over GF(11).
• a and b as 231 × 231 matrices over GF(11).
• a and b as 251 × 251 matrices over GF(11).
• A characteristic 0 representation.
• a and b as 22 × 22 matrices over Z.
• All primitive permutation representations
• a and b as permutations on 275 points.
• a and b as permutations on 2025 points.
• a and b as permutations on 2025 points- the image of the above under an outer automorphism.
• a and b as permutations on 7128 points.
• a and b as permutations on 15400 points - the cosets of 31+42S5.
• a and b as permutations on 15400 points - the cosets of 34M10.
• a and b as permutations on 22275 points - the cosets of L3(4).2.
• a and b as permutations on 22275 points - the cosets of 2A8.
• a and b as permutations on 22275 points - the cosets of 24A7.
• a and b as permutations on 22275 points - the cosets of the other 24A7.
• a and b as permutations on 113400 points - the cosets of M11.
• a and b as permutations on 299376 points.
The representations of 3.McL available are
• A and B as 126 × 126 matrices over GF(4).
• A and B as 396 × 396 matrices over GF(4).
• A and B as 42 × 42 matrices over GF(3).
• A and B as 45 × 45 matrices over GF(25).
• A and B as permutations on 66825 points - the cosets of 2.A8.
• A and B as permutations on 340200 points - the cosets of M11.
The representations of McL:2 available are
• c and d as 22 × 22 matrices over GF(2).
• c and d as 21 × 21 matrices over GF(3).
• c and d as 104 × 104 matrices over GF(3).
• c and d as 21 × 21 matrices over GF(5).
• c and d as 22 × 22 matrices over GF(7).
• c and d as 22 × 22 matrices over GF(11).
• c and d as permutations on 275 points.
• c and d as permutations on 4050 points.
• c and d as permutations on 7128 points.
• c and d as permutations on 22275 points - the cosets of L3(4).22.
• c and d as permutations on 44550 points.
The representations of 3.McL:2 available are
• C and D as 90 × 90 matrices over GF(5).
• C and D as 306 × 306 matrices over GF(5).
• C and D as 1278 × 1278 matrices over GF(5).
• C and D as 252 × 252 matrices over GF(4).

### 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 can be obtained by running this program on the standard generators. All conjugacy classes can therefore be obtained as suitable powers of these elements. Problems of algebraic conjugacy are not yet dealt with.