# ATLAS: Mathieu group M20 = 24:A5

Order = 960 = 26.3.5.
Mult = 42 × 2.
Out = S4.

### 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 2a.M20 = 21+4:A5 are preimages A and B where B has order 3, AB has order 5 and ABB has order 5.
Standard generators of the double cover 2b.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 2c.M20 are preimages A and B where B has order 3, AB has order 5, ABB has order 5 and AAB has order 3.

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...''.

### 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).
• a and b as permutations on 20 points - the natural representation.
• a and b as permutations on 20 points - not similar to the natural representation.
The representation of 2a.M20 = 21+4:A5 available is:
• A and B as permutations on 12 points - illustrating the embedding of this group in 2 wr PSL2(5).
The representation of 2b.M20 = 24:SL2(5) available is:
• A and B as permutations on 40 points - intransitive, with orbits 16 + 24.
• A and B as permutations on 160 points.
The representation of 2c.M20 available is:
• A and B as permutations on 24 points.