Mathieu group:M12

Definition
This group is defined as the Mathieu group of degree $$12$$. It is one of the five simple Mathieu groups. It is given by the following permutation representation, as a subgroup of symmetric group:S12 as follows. Consider the following set of size 12: the projective line over field:F11. Explicitly, this set can be written as:

$$\{ 0,1,2,3,4,5,6,7,8,9,10,\infty \}$$

We have projective special linear group:PSL(2,11), acting naturally on this set as fractional linear transformations by:

$$\begin{pmatrix} a & b \\ c & d \\\end{pmatrix} \mapsto \left(x \mapsto \frac{ax + b}{cx + d}\right)$$

This defines an embedding of $$PSL(2,11)$$ inside $$S_{12}$$.

The group $$M_{12}$$ is defined as the subgroup of $$S_{12}$$ generated by the image of $$PSL(2,11)$$ and the permutation given by $$x \mapsto 4x^2 - 3x^7$$, which as a permutation is:

$$\! (2,6,10,7)(3,9,4,5)$$

Note that this permutation fixes the points $$0,1,\infty$$.

Note further that since all generating permutations are even permutations, this group is in fact a subgroup of alternating group:A12.

GAP implementation
Unfortunately, GAP's SmallGroup library is not available for this order of group (95040) so the group cannot be constructed that way. It can be constructed in other ways:

The group is somewhat cumbersome to manipulate directly because of its large size. Information about the group, including its character table, can be accessed using the symbol "M12" -- see linear representation theory of Mathieu group:M12 for more.