Mathieu group M12 - Knowlpedia

The Mathieu group $M_{12}$ is the subgroup generated by the following permutations in the symmetric group $S_{12}$ :

\begin{aligned}a&=(1\,4)(3\,10)(5\,11)(6\,12),\\ b&=(1\,8\,9)(2\,3\,4)(5\,12\,11)(6\,10\,7). \end{aligned}

Thus $M_{12}=\langle a,b\rangle$ , 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_{12}|=95040=2^{6}\cdot3^{3}\cdot5\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 $12$ points is $5$ -transitive: any ordered $5$ -tuple of distinct points can be carried to any other such tuple. The element doing so is unique; this is called sharp transitivity.

The orbit of $\{1,2,3,4,5,9\}$ is a collection of $132$ blocks of size $6$ . Every $5$ -element subset lies in exactly one block, so these blocks form a Steiner system $S(5,6,12)$ . The generators preserve the system.

The natural point stabilizers connect the Mathieu groups.

References

ATLAS, Mathieu group M12, order heading.
ATLAS, permutation model M12G1-p12aB0, “Properties,” preserved Steiner system and “Download”; exact generators a and b.