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.
- 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