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 22 points is transitive on ordered 3-tuples
of distinct points and preserves the Steiner system 
. The full automorphism
group of this system contains 
as a subgroup of index
2.
The group is also the automorphism group of the two complementary strongly
regular graphs with parameters 
and 
(DistanceRegular.org).
The Schur multiplier of 
has group order 12, and
its outer automorphism group has group
order 2. According to the ATLAS, standard generators 
and 
may be chosen so that 
, 
, 
, 
, and the commutator ![[a,b]](/images/equations/MathieuGroupM22/Inline15.svg)
have orders
2, 4, 11, 11, and 6, respectively.
The group is implemented in the Wolfram Language as MathieuGroupM22[].
Pegg (2016) gives an interactive visualization of words in two group
generators for 
represented by matrices over a finite
field.
See also
Mathieu Group M23,
Mathieu Groups,
Simple Group,
Sporadic
Group,
Steiner System,
Strongly
Regular Graph
Explore with Wolfram|Alpha
References
Conway, J. H.; Curtis, R. T.; Norton, S. P.; Parker, R. A.; and Wilson, R. A. Atlas
of Finite Groups: Maximal Subgroups and Ordinary Characters for Simple Groups.
Oxford, England: Clarendon Press, 1985.Conway, J. H. and Sloane,
N. J. A. "The Golay Codes and the Mathieu Groups." Ch. 11
in Sphere
Packings, Lattices, and Groups, 2nd ed. New York: Springer-Verlag, pp. 299-330,
1993.DistanceRegular.org. "
." https://www.math.mun.ca/distanceregular/graphs/srg176.70.18.34.html.DistanceRegular.org.
"
."
https://www.math.mun.ca/distanceregular/graphs/srg176.105.68.54.html.
Pegg, E. Jr. "Sporadic Groups." Wolfram Demonstrations Project.
2016. https://demonstrations.wolfram.com/SporadicGroups/.Wilson,
R. A. "ATLAS: Mathieu Group 
." https://brauer.maths.qmul.ac.uk/Atlas/v3/scripts/group2.php?id=167&subpage=11.Referenced
on Wolfram|Alpha
Mathieu Group M22
Cite this as:
Weisstein, Eric W. "Mathieu Group M22."
From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/MathieuGroupM22.html
Subject classifications
