Incoming links: Mathieu group
en.wikipedia.org/w/index.php?title=Mathieu_group&oldid=1034060469#Multiply_transitive_groups is a nice characterization of 4 of the Mathieu groups.
TODO this would give a better motivation for the Mathieu group
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 is a permutation group on elements. There isn't an obvious algorithmic relationship between 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 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.
Mathieu group section of Why Do Sporadic Groups Exist? by Another Roof (2023)
Source. Only discusses Mathieu group but is very good at that.