The five Mathieu groups 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 MathieuGroupM24
All the sporadic Mathieu groups are multiply transitive . The following table summarizes some properties of the Mathieu groups, where
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 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." 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 https://mathworld.wolfram.com/MathieuGroups.html
Subject classifications 
, 
, 
, 
, 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 
. 
indicates the transitivity and 
is the length of the minimal permutation support (from which the groups derive their designations). 
