  We investigate the idea that the double cover of the rotational icosahedral
symmetry group is the family symmetry group in the quark sector. The
icosahedral (A5) group was previously proposed as a viable family symmetry
group for the leptons. To incorporate the quarks, it is highly advantageous to
extend the group to its double cover, as in the case of tetrahedral (A4)
symmetry. We provide the basic group theoretical tools for flavor
model-building based on the binary icosahedral group I' and construct a model
of the quark masses and mixings that yields many of the successful predictions
of the well-known U(2) quark texture models.
