  A $k$-orbit maniplex is one that has $k$ orbits of flags under the action of
its automorphism group. In this paper we extend the notion of symmetry type
graphs of maps to that of maniplexes and polytopes and make use of them to
study $k$-orbit maniplexes, as well as fully-transitive 3-maniplexes. In
particular, we show that there are no fully-transtive $k$-orbit 3-mainplexes
with $k > 1$ an odd number, we classify 3-orbit mainplexes and determine all
face transitivities for 3- and 4-orbit maniplexes. Moreover, we give generators
of the automorphism group of a polytope or a maniplex, given its symmetry type
graph. Finally, we extend these notions to oriented polytopes, in particular we
classify oriented 2-orbit maniplexes and give generators for their orientation
preserving automorphism group.
