  The goal of this paper is to study primitive groups that are contained in the
union of maximal (in the symmetric group) imprimitive groups. The study of
types of permutations that appear inside primitive groups goes back to the
origins of the theory of permutation groups. However, this is another instance
of a situation common in mathematics in which a very natural problem turns out
to be extremely difficult. Fortunately, the enormous progresses of the last few
decades seem to allow a new momentum on the attack to this problem. In this
paper we prove that there are infinite families of primitive groups contained
in the union of imprimitive groups and propose a new hierarchy for primitive
groups based on that fact. In addition to the previous results and hierarchy,
we introduce some algorithms to handle permutations, provide the corresponding
GAP implementation, solve some open problems, and propose a large list of open
problems.
