  Given a compact Riemann surface $X$ with an action of a finite group $G$, the
group algebra Q[G] provides an isogenous decomposition of its Jacobian variety
$JX$, known as the group algebra decomposition of $JX$. We consider the set of
equisymmetric Riemann surfaces $\mathcal{M}(2n-1, D_{2n}, \theta)$ for all
$n\geq 2$. We study the group algebra decomposition of the Jacobian $JX$ of
every curve $X\in \mathcal{M}(2n-1, D_{2n},\theta)$ for all admissible actions,
and we provide affine models for them. We use the topological equivalence of
actions on the curves to obtain facts regarding its Jacobians. We describe some
of the factors of $JX$ as Jacobian (or Prym) varieties of intermediate
coverings. Finally, we compute the dimension of the corresponding Shimura
domains.
