The five Mathieu groups ,
, , , and were the first sporadic
groups discovered, having been found in 1861 and 1873 by Mathieu. Frobenius showed
that all the Mathieu groups are subgroups of .

The sporadic Mathieu groups are implemented in the Wolfram Language as MathieuGroupM11 [],
MathieuGroupM12 [],
MathieuGroupM22 [],
MathieuGroupM23 [],
and MathieuGroupM24 [].

All the sporadic Mathieu groups are multiply transitive . The following table summarizes some properties of the Mathieu groups, where indicates the transitivity and is the length of the minimal permutation support (from which
the groups derive their designations).

group order factorization 4 11 7920 5 12 95040 3 22 443520 4 23 10200960 5 24 244823040

The Mathieu groups are most simply defined as automorphism groups of Steiner systems , as summarized in
the following table.

Mathieu group Steiner system

See also Automorphism Group ,

Large Witt Graph ,

Simple Group ,

Sporadic
Group ,

Steiner System ,

Transitive
Group ,

Witt Geometry
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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. Dixon, J. and Mortimer, B. Permutation
Groups. New York: Springer-Verlag, 1996. Rotman, J. J. Ch. 9
in An
Introduction to the Theory of Groups, 4th ed. New York: Springer-Verlag,
1995. Wilson, R. A. "ATLAS of Finite Group Representation."
http://brauer.maths.qmul.ac.uk/Atlas/v3/spor/ . Referenced
on Wolfram|Alpha Mathieu Groups
Cite this as:
Weisstein, Eric W. "Mathieu Groups." From
MathWorld --A Wolfram Web Resource. https://mathworld.wolfram.com/MathieuGroups.html

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