Mathieu group M24 - Knowlpedia

The Mathieu group $M_{24}$ is the subgroup generated by the following permutations in the symmetric group $S_{24}$ :

\begin{aligned}a&=(1\,4)(2\,7)(3\,17)(5\,13)\\ &\quad{}(6\,9)(8\,15)(10\,19)(11\,18)\\ &\quad{}(12\,21)(14\,16)(20\,24)(22\,23),\\ b&=(1\,4\,6)(2\,21\,14)(3\,9\,15)(5\,18\,10)\\ &\quad{}(13\,17\,16)(19\,24\,23). \end{aligned}

Thus $M_{24}=\langle a,b\rangle$ , with permutation composition as the group operation. Cycles list successive images, unlisted points are fixed, and products act from right to left.

Order and structure

This is a nonabelian finite simple group, with

|M_{24}|=244823040=2^{10}\cdot3^{3}\cdot5\cdot7\cdot11\cdot23.

It belongs to the Mathieu display block of the sporadic groups. This grouping records the ATLAS organization, without asserting a direct embedding into another displayed group.

The natural permutation action

The action on $24$ points is $5$ -transitive: any ordered $5$ -tuple of distinct points can be carried to any other such tuple.

The orbit of $\{1,2,3,4,5,11,17,24\}$ is a collection of $759$ blocks of size $8$ . Every $5$ -element subset lies in exactly one block, so these blocks form a Steiner system $S(5,8,24)$ . The generators preserve the system.

The natural point stabilizers connect the Mathieu groups.

References

ATLAS, Mathieu group M24, order heading.
ATLAS, permutation model M24G1-p24B0, “Properties,” preserved Steiner system and “Download”; exact generators a and b.