The five Mathieu groups M 11 M 12 M 22 M 23 M 24 sporadic groups discovered, having
been found in 1861 and 1873 by Mathieu. Frobenius showed that all the Mathieu groups
are subgroups of M 24
The sporadic Mathieu groups are implemented in the Wolfram Language as MathieuGroupM11 MathieuGroupM12 MathieuGroupM22 MathieuGroupM23 MathieuGroupM24
All the sporadic Mathieu groups are multiply transitive . The following table summarizes some properties of the Mathieu groups, where
The natural group actions of the Mathieu groups preserve the Steiner systems summarized in the following
table.
The groups M 11 M 12 M 23 M 24 automorphism groups of their corresponding
systems. The full automorphism group of M 22 subgroup of index
2.
See also Automorphism Group ,
Large Witt Graph ,
Mathieu Group M11 ,
Mathieu
Group M12 ,
Mathieu Group M22 ,
Mathieu
Group M23 ,
Mathieu Group M24 ,
Simple
Group ,
Sporadic Group ,
Steiner
System ,
Transitive Group
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. 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. Dixon, J. and Mortimer, B. Permutation
Groups. Rotman, J. J. Ch. 9
in An
Introduction to the Theory of Groups, 4th ed. Wilson, R. A. "ATLAS of Finite Group Representation."
https://brauer.maths.qmul.ac.uk/Atlas/v3/spor/ . Referenced
on Wolfram|Alpha Mathieu Groups
Cite this as:
Weisstein, Eric W. "Mathieu Groups." From
MathWorld https://mathworld.wolfram.com/MathieuGroups.html
Subject classifications
indicates the transitivity and 
is the length of the minimal permutation support (from which
the groups derive their designations).
contains M22
as a subgroup of index
2.
