Building SO${}_{10}$- models with ${\mathbb{D}}_4$ symmetry

Using characters of finite group representations and monodromy of matter curves in F-GUT, we complete partial results in literature by building SO$% {10}$modelswithdihedral$$\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 $S{O}_{10}\times {\mathbb{D}}_4$ 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.