ATLAS: Mathieu group M_{20} = 2^{4}:A_{5}
Order = 960 = 2^{6}.3.5.
Mult = 4^{2} × 2.
Out = S_{4}.
The page for the image A_{5} is available here.
The following information is available for covers of M_{20} = 2^{4}:A_{5}:
Standard generators of the Mathieu group M_{20} = 2^{4}:A_{5}
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 A_{5}.
Standard generators of the double cover 2_{1}.M_{20} = 2^{1+4}:A_{5}
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 2_{2}.M_{20} = 2^{4}:SL_{2}(5)
are preimages A and B where B has order 3 and
AB has order 5.
Standard generators of the double cover 2_{3}.M_{20} 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 4_{1}.M_{20}
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 4_{2}.M_{20}
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 2^{2}.M_{20} are preimages A and B where B has order 3 and AB has order 5, and where the preimages map onto standard generators of 2_{1}.M_{20}.
There are just two isomorphism classes of covers 4.M_{20}, and these both map onto 2_{1}.M_{20}. There are 6 covers of each isomorphism type. We have named them so that 4_{1}.L_{3}(4) contains 4_{1}.M_{20} and 4_{2}.L_{3}(4) contains 4_{2}.M_{20}.
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.
A presentation for M_{20} on its standard generators is given below.
< a, b  a^{4} = b^{3} = (ab)^{5} = (ab^{1})^{5} = (a^{2}b)^{3} = (abab^{1}ab^{1}a^{1}b^{1})^{2} = 1 >.
Without the last relation, we get a presentation for 2c.M_{20}. [Lengths are 52 and 36 respectively.]
The representations of M_{20} = 2^{4}:A_{5} available are:
 a and
b as
permutations on 16 points  primitive (2transitive 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 2_{1}.M_{20} = 2^{1+4}:A_{5} (the 2^{1+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 2_{2}.M_{20} = 2^{4}:SL_{2}(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 2_{3}.M_{20} available is:

Permutations on 24 points:
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
The representations of 4_{2}.M_{20} available are:

Dimension 4 over Z[i]:
A and B (Magma).

Dimension 4 over Z[i] (different basis):
A and B (Magma).
Representatives of the 9 conjugacy classes of M_{20} =
2^{4}:A_{5} can be taken to be as follows:
 1A: identity [or a^{4}].
 2A: a^{2}.
 2B: ababa^{1}bab^{1}.
 4A: a.^{ }
 4B: aba^{2}b^{1}.
 4C: ababa^{1}ba^{1}b^{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.)
Go to main ATLAS (version 2.0) page.
Go to miscellaneous groups page.
Go to old M20 page  ATLAS version 1.
Anonymous ftp access is also available.
See here for details.
Version 2.0 created on 4th March 2002.
Last updated 06.03.02 by JNB.
Information checked to
Level 0 on 04.03.02 by JNB.
R.A.Wilson, R.A.Parker and J.N.Bray.