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 $\text{D}(G)$ 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 $p\mid q-1$ and let $C_q⋊C_p$ denote the non-abelian group of order $pq$. It is known that $\text{D}(C_q⋊C_p)=2q$. In this paper, we describe all minimal product-one sequences of length $2q$ over $C_q⋊C_p$. 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.