The Schur multiplier and the outer automorphism group are both trivial.
This action is the automorphism group of a Steiner system S(4,5,11).
The permutation representations on 11 and 12 points can both be seen inside the Mathieu group M12 as two different embeddings of M11 in M12, exchanged by an outer automorphism.
These have the smallest dimension of any faithful linear representations of M11 over any field.
There are 5 conjugacy classes of maximal subgroups of M11 as follows: The maximum order of any element in M11 is 11.