The transitivity of Conwayʹs image

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