The Mathieu group 
is a sporadic group of group
order 
.
It is a simple group and a subgroup
of the symmetric group 
. Its natural group action
on 24 points is transitive on ordered 5-tuples
of distinct points. It is the automorphism group
of the Steiner system 
and of the extended binary Golay
code.
The group is also the automorphism group of the large Witt graph (Brouwer).
The Schur multiplier and outer automorphism group of 
both have group order
1. The Mathieu group M23 is
a subgroup of index
24. According to the ATLAS, standard generators 
and 
may be chosen so that 
, 
, 
, 
, and 
have orders 2, 3, 23,
4, and 12, respectively.
The group is implemented in the Wolfram Language as MathieuGroupM24[].
Pegg (2016) gives an interactive visualization of words in two group
generators for 
represented by matrices over a finite
field.
Pegg, E. Jr. "Sporadic Groups."
Wolfram Demonstrations Project. 2016. 
."