  At special loci in their moduli spaces, Calabi-Yau manifolds are endowed with
discrete symmetries. Over the years, such spaces have been intensely studied
and have found a variety of important applications. As string compactifications
they are phenomenologically favored, and considerably simplify many important
calculations. Mathematically, they provided the framework for the first
construction of mirror manifolds, and the resulting rational curve counts.
Thus, it is of significant interest to investigate such manifolds further. In
this paper, we consider several unexplored loci within familiar families of
Calabi-Yau hypersurfaces that have large but unexpected discrete symmetry
groups. By deriving, correcting, and generalizing a technique similar to that
of Candelas, de la Ossa and Rodriguez-Villegas, we find a calculationally
tractable means of finding the Picard-Fuchs equations satisfied by the periods
of all 3-forms in these families. To provide a modest point of comparison, we
then briefly investigate the relation between the size of the symmetry group
along these loci and the number of nonzero Yukawa couplings. We include an
introductory exposition of the mathematics involved, intended to be accessible
to physicists, in order to make the discussion self-contained.
