  In [1] it has been shown that the Cabibbo angle theta_C might arise from a
dihedral flavor symmetry which is broken to different (directions of) subgroups
in the up and the down quark sector. This leads to a prediction of theta_C in
terms of group theoretical quantities only, i.e. the index n of the dihedral
group D_n, the index j of the fermion representation 2_j and the preserved
subgroups indicated by m_u and m_d. Here we construct a low energy model which
incorporates this idea. The gauge group is the one of the Standard Model and
D_7 x Z_2 ^(aux) serves as flavor symmetry. The additional Z_2 ^(aux) is
necessary in order to maintain two sets of Higgs fields, one which couples only
to up quarks and another one coupling only to down quarks. We assume that D_7
is broken spontaneously at the electroweak scale by vacuum expectation values
of SU(2)_L doublet Higgs fields. The quark masses and mixing parameters can be
accommodated well. Furthermore, the potential of the Higgs fields is studied
numerically in order to show that the required configuration of the vacuum
expectation values can be achieved. We also comment on more minimalist models
which explain the Cabibbo angle in terms of group theoretical quantities, while
theta_{13}^q and theta_{23}^q vanish at leading order. Finally, we perform a
detailed numerical study of the lepton mixing matrix V_{MNS} in which one of
its elements is entirely determined by the group theory of a dihedral symmetry.
Thereby, we show that nearly tri-bi-maximal mixing can also be produced by a
dihedral flavor group with preserved subgroups.
