  Motivated by the Babai conjecture and the Cerny conjecture, we study the
reset thresholds of automata with the transition monoid equal to the full
monoid of transformations of the state set. For automata with $n$ states in
this class, we prove that the reset thresholds are upper-bounded by $2n^2-6n+5$
and can attain the value $\tfrac{n(n-1)}{2}$. In addition, we study diameters
of the pair digraphs of permutation automata and construct $n$-state
permutation automata with diameter $\tfrac{n^2}{4} + o(n^2)$.
