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 12 points is transitive on ordered 5-tuples
of distinct points, with a unique group element
sending any such tuple to any other. It is the automorphism
group of the Steiner system 
.
The Schur multiplier and outer automorphism group of 
both have group order
2. A point stabilizer in its natural group
action is isomorphic to M11.
According to the ATLAS, standard generators 
and 
may be chosen so that 
, 
, 
, and 
have orders 2, 3, 11,
and 6, respectively.
The group is implemented in the Wolfram Language as MathieuGroupM12[].
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. 
."