Building SO10-models with D4 symmetry

Using characters of finite group representations and monodromy of matter curves in F-GUT, we complete partial results in literature by building SO10 models with dihedral D4 discrete symmetry. We first revisit the S4-and S3-models from the discrete group character view; then we extend the construction to D4. We find that there are three types of SO10×D4 models depending on the ways the S4-triplets break down in terms of irreducible D4-representations: (α) as 1+,−⊕1+,−⊕1−,+; or (β)1+,+⊕1+,−⊕1−,−; or also (γ) 1+,−⊕20,0. Superpotentials and other features are also given.