Building SO$_{10}$- models with $\mathbb{D}_{4}$ symmetry

Abstract: Using characters of finite group representations and monodromy of matter curves in F-GUT, we complete partial results in literature by building SO$% {10}$ models with dihedral $\mathbb{D}{4}$ discrete symmetry. We first revisit the $\mathbb{S}{4}$-and $\mathbb{S}{3}$-models from the discrete group character view, then we extend the construction to $\mathbb{D}{4}$.\ We find that there are three types of $SO{10}\times \mathbb{D}{4}$ models depending on the ways the $\mathbb{S}{4}$-triplets break down in terms of irreducible $\mathbb{D}{4}$- representations: $\left({\alpha} \right) $ as $\boldsymbol{1}{{+,-}}\oplus \boldsymbol{1}{{+,-}}\oplus \boldsymbol{1}{{-,+}};$ or $\left({\beta}\right) \boldsymbol{\ 1}{{+,+}}\oplus \boldsymbol{1}{{+,-}}\oplus \boldsymbol{1}{{-,-}};$ or also $\left({\gamma}\right) $ $\mathbf{1}{{+,-}}\oplus \mathbf{2}{_{0,0}}$. Superpotentials and other features are also given.