Mathieu group M22 - Knowlpedia

The Mathieu group $M_{22}$ is the subgroup generated by the following permutations in the symmetric group $S_{22}$ :

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

Thus $M_{22}=\langle a,b\rangle$ , 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_{22}|=443520=2^{7}\cdot3^{2}\cdot5\cdot7\cdot11.

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 $22$ points is $3$ -transitive: any ordered $3$ -tuple of distinct points can be carried to any other such tuple.

The orbit of $\{1,2,3,10,18,20\}$ is a collection of $77$ blocks of size $6$ . Every $3$ -element subset lies in exactly one block, so these blocks form a Steiner system $S(3,6,22)$ . The generators preserve the system.

The natural point stabilizers connect the Mathieu groups.

References

ATLAS, Mathieu group M22, order heading.
ATLAS, permutation model M22G1-p22B0, “Properties,” preserved Steiner system and “Download”; exact generators a and b.