# ATLAS: Mathieu group M20 = 24:A5

Order = 960 = 26.3.5.
Mult = 42 × 2.
Out = S4.
The page for the image A5 is available here.

The following information is available for covers of M20 = 24:A5:

### Standard generators

Standard generators of the Mathieu group M20 = 24:A5 are a and b where a has order 4, b has order 3 and ab has order 5. These generators map onto standard generators of A5.
Standard generators of the double cover 21.M20 = 21+4:A5 are preimages A and B where B has order 3, AB has order 5, ABB has order 10 and AAB has order 6.
Standard generators of the double cover 22.M20 = 24:SL2(5) are preimages A and B where B has order 3 and AB has order 5.
Standard generators of the double cover 23.M20 are preimages A and B where B has order 3, AB has order 5, ABB has order 5 and AAB has order 3.

We haven't fully checked that the standard generators for 4.M20 and 2^2.M20 are sufficient to define them up to automorphisms.
Standard generators of the fourfold cover 41.M20 are preimages A and B where B has order 3, AB has order 5, A has order 4 and ABABAAABABB has order 2.
Standard generators of the fourfold cover 42.M20 are preimages A and B where B has order 3, AB has order 5, A has order 4 and ABABAAABABB has order 2.
Standard generators for any 22.M20 are preimages A and B where B has order 3 and AB has order 5, and where the preimages map onto standard generators of 21.M20.

There are just two isomorphism classes of covers 4.M20, and these both map onto 21.M20. There are 6 covers of each isomorphism type. We have named them so that 41.L3(4) contains 41.M20 and 42.L3(4) contains 42.M20.

NB: It is possible that we may change some of the definitions of standard generators of the covers of M20 above when we come to define standard generators for all the covers of M20. These changes will be subtle, and the definition will still have the condition ``...preimages A and B where B has order 3 and AB has order 5...''.

NB: We have altered some definitions of standard generators since version 1. We reserve the right to alter the definitions further without giving any notice.

NB: Representation order NOT fixed and liable to change without notice.

### Presentations

A presentation for M20 on its standard generators is given below.

< a, b | a4 = b3 = (ab)5 = (ab-1)5 = (a2b)3 = (abab-1ab-1a-1b-1)2 = 1 >.

Without the last relation, we get a presentation for 2c.M20. [Lengths are 52 and 36 respectively.]

### Representations

The representations of M20 = 24:A5 available are:
• a and b as permutations on 16 points - primitive (2-transitive in fact).
• Permutations on 20a points - the cosets of 2^4:3 (`natural'): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Permutations on 20b points - the cosets of 4A^2:3: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Permutations on 20c points - the cosets of 4B^2:3: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Permutations on 20d points - the cosets of 4C^2:3: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
The representations of 21.M20 = 21+4:A5 (the 21+4 being abelian) available are:
• 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 20a points: A and B (Meataxe), A and B (Meataxe binary), A and B (GAP).
• Permutations on 20b points: A and B (Meataxe), A and B (Meataxe binary), A and B (GAP).
• All faithful irreducibles in characteristic 0.
• Dimension 6a over Z - monomial: A and B (Magma).
• Dimension 6b over Z - monomial: A and B (Magma).
• Dimension 10a over Z - monomial: A and B (Magma).
• Dimension 10b over Z - monomial: A and B (Magma).
• Dimension 12a over Z[b5]: A and B (Magma).
• Dimension 12b over Z[b5]: A and B (Magma).
• Dimension 12b over Z[b5] (different basis): A and B (Magma).
• Dimension 20 over Z: A and B (Magma).
• Dimension 24 over Z - reducible over Q(b5): A and B (Magma).
The representation of 22.M20 = 24:SL2(5) available is:
• A and B as permutations on 40 points - intransitive, with orbits 16 + 24.
• Permutations on 120 points: A and B (Meataxe), A and B (Meataxe binary), A and B (GAP).
• A and B as permutations on 160 points.
The representation of 23.M20 available is:
The representations of 42.M20 available are:
• Dimension 4 over Z[i]: A and B (Magma).
• Dimension 4 over Z[i] (different basis): A and B (Magma).

### Conjugacy classes

Representatives of the 9 conjugacy classes of M20 = 24:A5 can be taken to be as follows:
• 1A: identity [or a4].
• 2A: a2.
• 2B: ababa-1bab-1.
• 4A: a.
• 4B: aba2b-1.
• 4C: ababa-1ba-1b-1.
• 3A: b.
• 5A: ab.
• 5B: abab = (ab)2.

This below is not quite true!!!
In 2a.M20, the given class representatives are class +Xm whenever class Xm splits in 2a.M20. (Actually, there are outer automorphisms interchanging some of the +Xm with -Xm, so it is not so surprising that all classes `become' +Xm in the double cover.)

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Version 2.0 created on 4th March 2002.
Last updated 11.04.05 by RAW.
Information checked to Level 0 on 04.03.02 by JNB.
R.A.Wilson, R.A.Parker and J.N.Bray.