  The largest group which occurs as the rotational symmetries of a
three-dimensional reflexive polytope is the symmetric group on four elements.
There are three pairs of three-dimensional reflexive polytopes with this
symmetry group, up to isomorphism. We identify a natural one-parameter family
of K3 surfaces corresponding to each of these pairs, show that the symmetric
group on four elements acts symplectically on members of these families, and
show that a general K3 surface in each family has Picard rank 19. The
properties of two of these families have been analyzed in the literature using
other methods. We compute the Picard-Fuchs equation for the third Picard rank
19 family by extending the Griffiths-Dwork technique for computing Picard-Fuchs
equations to the case of semi-ample hypersurfaces in toric varieties. The
holomorphic solutions to our Picard-Fuchs equation exhibit modularity
properties known as "Mirror Moonshine"; we relate these properties to the
geometric structure of our family.
