  Recent work from authors across disciplines has made substantial
contributions to counting rules (Maxwell type theorems) which predict when an
infinite periodic structure would be rigid or flexible while preserving the
periodic pattern, as an engineering type framework, or equivalently, as an
idealized molecular framework. Other work has shown that for finite frameworks,
introducing symmetry modifies the previous general counts, and under some
circumstances this symmetrized Maxwell type count can predict added finite
flexibility in the structure.
  In this paper we combine these approaches to present new Maxwell type counts
for the columns and rows of a modified orbit matrix for structures that have
both a periodic structure and additional symmetry within the periodic cells. In
a number of cases, this count for the combined group of symmetry operations
demonstrates there is added finite flexibility in what would have been rigid
when realized without the symmetry. Given that many crystal structures have
these added symmetries, and that their flexibility may be key to their physical
and chemical properties, we present a summary of the results as a way to
generate further developments of both a practical and theoretic interest.
