Contains the first sporadic groups discovered by far: 11 and 12 in 1861, and 22, 23 and 24 in 1973. And therefore presumably the simplest! The next sporadic ones discovered were the Janko groups, only in 1965!

Each $M_{n}$ is a permutation group on $n$ elements. There isn't an obvious algorithmic relationship between $n$ and the actual group.

TODO initial motivation? Why did Mathieu care about k-transitive groups?

Their; k-transitive group properties seem to be the main characterization, according to Wikipedia:Looking at the classification of k-transitive groups we see that the Mathieu groups are the only families of 4 and 5 transitive groups other than symmetric groups and alternating groups. 3-transitive is not as nice, so let's just say it is the stabilizer of $M_{2}3$ and be done with it.

- 22 is 3-transitive but not 4-transitive.
- four of them (11, 12, 23 and 24) are the only sporadic 4-transitive groups as per the classification of 4-transitive groups (no known simpler proof as of 2021), which sounds like a reasonable characterization. Note that 12 and 25 are also 5 transitive.

- Sporadic group | 8, 482, 17
- Classification of finite simple groups | 186, 949, 23
- Classification of finite groups | 69, 1k, 27
- Normal subgroup | 149, 1k, 30
- Quotient group | 51, 2k, 33
- Subgroup | 0, 2k, 35
- Group | 0, 5k, 89
- Algebra | 0, 8k, 171
- Mathematics | 17, 28k, 633
- Ciro Santilli's Homepage | 262, 218k, 4k

- Classification of 4-transitive groups | 19
- Classification of k-transitive groups | 22, 155, 6