The point stabilizer $H$ of the Mathieu group $M_{11}$ in its $2$-transitive action on $12$ points acts transitively in the doubly transitive action of $M_{11}$ on $11$ points. This follows because $11$ divides $\lvert H\rvert$. (Or from the fact that the scalar product of the two permutation characters of degrees $11$ and $12$ is $1$.)
Peter Mueller
- 22.5k
- 1
- 75
- 107