Mathieu Groups
The first Simple Sporadic Groups discovered. 
, 
, 
,
, 
were discovered in 1861 and 1873 by Mathieu. Frobenius showed that all the Mathieu groups areSubgroups of .
The Mathieu groups are most simply defined as Automorphism groups of Steiner Systems. corresponds to 
and corresponds to 
. and are TransitivePermutation Groups of 11 and 23 elements.
The Orders of the Mathieu groups are
See also Sporadic Group
References
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. 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://for.mat.bham.ac.uk/atlas/html/contents.html#spo.
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