Building SO10-models with D4D4 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 D4D4 discrete symmetry. We first revisit the S4S4-and S3S3-models from the discrete group character view; then we extend the construction to D4D4. We find that there are three types of SO10×D4SO10×D4 models depending on the ways the S4S4-triplets break down in terms of irreducible D4D4-representations: (α  ) as 1+,−⊕1+,−⊕1−,+1+,−⊕1+,−⊕1−,+; or (β)1+,+⊕1+,−⊕1−,−(β)1+,+⊕1+,−⊕1−,−; or also (γ  ) 1+,−⊕20,01+,−⊕20,0. Superpotentials and other features are also given.