The Mathieu groups, denoted , , , and are sporadic finite simple groups. They were first described in the 1860-70s by Émile Mathieu, and the first such groups to be discovered.
The orders of the groups are as follows:
The Matthieu group is the automorphism group of the binary Golay code; this is a vector space over the field . The other groups can be obtained as stabilisers of various (sets of) elements of the Golay code, and hence are subgroups of . The Mathieu groups form the so-called first generation of the happy family: the collection of 20 sporadic groups which are subgroups of the Monster group.