  We give a new characterization of primitive permutation groups tied to the
notion of completely reachable automata. Also, we introduce sync-maximal
permutation groups tied to the state complexity of the set of synchronizing
words of certain associated automata and show that they are contained between
the $2$-homogeneous and the primitive groups. Lastly, we define $k$-reachable
groups in analogy with synchronizing groups and motivated by our
characterization of primitive permutation groups. But the results show that a
$k$-reachable permutation group of degree $n$ with $6 \le k \le n - 6$ is
either the alternating or the symmetric group.
