The five Mathieu groups M11, M12, M22, M23, and M24 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 M24.
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 |  |  | group order | prime factorization |
| M11 | 4 | 11 | 7920 |  |
| M12 | 5 | 12 | 95040 |  |
| M22 | 3 | 22 | 443520 |  |
| M23 | 4 | 23 | 10200960 |  |
| M24 | 5 | 24 | 244823040 |  |
The natural group actions of the Mathieu groups preserve the Steiner systems summarized in the following table.
The groups M11, M12, M23, and M24
are the full automorphism groups of their corresponding
systems. The full automorphism group of 
contains M22
as a subgroup of index
2.
