  The directed Cayley diameter of a finite group is investigated in terms of
the monoid of product-one sequences over the group, via the new notion of
directed geodesic atoms. Two quantities associated to the set of directed
geodesic atoms provide lower and upper bounds for the directed Cayley diameter.
An algorithm for computing the directed geodesic atoms is implemented in GAP,
and is applied to determine the above mentioned quantities for all non-abelian
groups of order at most $42$, and for the alternating group of degree $5$.
Furthermore, the small and large Davenport constants of all these groups are
computed (excepting the large Davenport constant for $A_5$), extending thereby
the formerly obtained results on the groups of order less than $32$. Along the
way the directed Cayley diameter of a finite abelian group is expressed in
terms of its invariants.
