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.