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In the four-dimensional effective theory from string compactificationdiscrete flavor symmetries arise from symmetric structure of the compactifiedspace and generally contain both the $R$ symmetry and non-$R$ symmetry. Wepoint out that a new type of non-Abelian flavor symmetry can also appear if thecompact space is non- commutative. Introducing the dihedral group $D_4$ as sucha new type of flavor symmetry together with the $R$ symmetry and non-$R$symmetry in $SU(6) \times SU(2)_R$ model, we explain not only fermion masshierarchies but also hierarchical energy scales including the breaking scale ofthe GUT-type gauge symmetry, intermediate Majorana masses of R-handed neutrinosand the scale of $\mu$-term.Comment: 28pages, Latex, no figures, Revised version includes some detail

Flavor Symmetry on Non-Commutative Compact Space and SU(6) x SU(2)R Model