Mathieu group:M10

Definition
This group, denoted $$M_{10}$$, is the member of family::Mathieu group of degree $$10$$. Explicitly, it can be defined as the isotropy subgroup (stabilizer) of two points under the natural action of Mathieu group:M12 on the projective line over field:F11. Since the action is a 2-transitive group action, it does not matter which two points we choose, but for concreteness, we can choose $$0$$ and $$\infty$$.

Although it is one of the Mathieu groups, it is not among the five sporadic simple Mathieu groups. Rather, it is among the two non-simple Mathieu groups. The other is Mathieu group:M9. The Mathieu group $$M_{21}$$ is isomorphic to projective special linear group:PSL(3,4) and is hence simple but not a sporadic simple group.