On Dedekind domains whose class groups are direct sums of cyclic groups

For a given family $(G_i{)}_{i\in \mathbb{N}}$ of finitely generated abelian groups, we construct a Dedekind domain $D$ having the following properties. \begin{enumerate} \item $\text{Pic}(D)\cong {⨁}_{i\in \mathbb{N}}{G}_i$. \item For each $i\in \mathbb{N}$, there exists a submonoid $S_i\subseteq D^{\bullet }$ with $\text{Pic}(D_{S_i})\cong G_i$. \item Each class of $\text{Pic}(D)$ and of all $\text{Pic}(D_{S_i})$ contains infinitely many prime ideals. \end{enumerate} Furthermore, we study orders as well as sets of lengths in the Dedekind domain $D$ and in all its localizations $D_{S_i}$.