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The directed Cayley diameter of a finite group is investigated using themonoid of product-one sequences over the group. This relation is used todevelop an algorithm for computing bounds for the directed Cayley diameter of anon-abelian finite group, based on a method to compute the so-called geodesicatoms in the monoid of product-one sequences. The algorithm has beenimplemented in GAP, and was applied to determine the directed Cayley diameterfor several non-abelian groups of order at most $39$. Furthermore, the smalland large Davenport constants of all non-abelian groups of order at most $39$are computed, extending thereby the formerly obtained results on the groups oforder less than $32$. Along the way the directed Cayley diameter of a finiteabelian group is expressed in terms of its invariants.

The directed Cayley diameter and the Davenport constant