The transitivity of Conway's M13M13

J.H. Conway introduced a set of permutations called M13M13 by a game related to the projective plane of order 3. The set M13M13 consists of certain permutations on 13 letters, and contains the Mathieu group M12M12. W.J. Martin and B.E. Sagan generalized the concept of transitivity for a set of permutations by defining λλ-transitivity for each partition λλ of the degree of the permutations. We determine the partitions λλ of 13 for which the set M13M13 is λλ-transitive.