On minimal product-one sequences of maximal length over the non-abelian group of order $pq$

Let G be a finite group. A sequence over G is a finite multiset of elements of G, and it is called product-one if its terms can be ordered so that their product is the identity of G. The large Davenport constant (G) is the maximal length of a minimal product-one sequence, that is, a product-one sequence that cannot be partitioned into two nontrivial product-one subsequences. Let $p,q$ be odd prime numbers with p q-1 and let Cq Cₚ denote the non-abelian group of order $pq$. It is known that (Cq Cₚ) = 2q. In this paper, we describe all minimal product-one sequences of length $2q$ over Cq Cₚ. As an application, we further investigate the k-th elasticity (and, consequently, the union of sets containing k) of the monoid of product-one sequences over these groups.