The Mathieu group 
is a sporadic group of group
order 
.
It is a simple group and a subgroup
of the symmetric group 
. Its group action on
23 points is transitive on ordered 4-tuples
of distinct points. It is the sixth-smallest sporadic
group and the automorphism group of the
Steiner system 
.
The group is also the automorphism group of the truncated Witt graph and
of the strongly regular graph with parameters
(DistanceRegular.org).
The following three permutations form a representative triple of elements in conjugacy classes of types
, 
, and 
used by Huang et al. (2026). Using the authors' convention
that 
acts on the left, they satisfy 
and generate 
.
 |  |  |
(1)
|
 |  |  |
(2)
|
 |  |  |
(3)
|
The Mathieu group 
was the last sporadic group not known to occur
as a Galois group over 
. Huang et al. (2026) proved the stronger result that
there is a regular Galois extension of
with Galois
group 
.
The splitting field of the following degree-23
polynomial is an explicit 
-extension of 
whose ramification occurs
only at the primes 2, 3, and 23.
 |
(4)
|
The group is implemented in the Wolfram Language as MathieuGroupM23[].
Pegg (2016) gives an interactive visualization of words in two group
generators for 
represented by matrices over a finite
field, rather than by the degree-23 permutations
above.
Graph." 
Is a Galois Group over 
." Aug. 8, 2026. 
Pegg, E. Jr. "Sporadic Groups." Wolfram Demonstrations Project.
2016.