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 D 4 discrete symmetry. We first revisit the S 4 -and S 3 -models from the discrete group character view; then we extend the construction to D 4 . We find that there are three types of SO 10 × D 4 models depending on the ways the S 4 -triplets break down in terms of irreducible D 4 -representations: (α) as 1 +,− ⊕ 1 +,− ⊕ 1 −,+ ; or (β) 1 +,+ ⊕1 +,− ⊕1 −,− ; or also (γ) 1 +,− ⊕2 0,0 . Superpotentials and other features are also given.