  A large class of orbifold quiver gauge theories admits the action of finite
Heisenberg groups of the form \prod_i Heis(Z_{q_i} x Z_{q_i}). For an Abelian
orbifold generated by \Gamma, the Z_{q_i} shift generator in each Heisenberg
group is one cyclic factor of the Abelian group \Gamma. For general non-Abelian
\Gamma, however, we find that the shift generators are the cyclic factors in
the Abelianization of \Gamma. We explicitly show this for the case
\Gamma=\Delta(27), where we construct the finite Heisenberg group symmetries of
the field theory. These symmetries are dual to brane number operators counting
branes on homological torsion cycles, which therefore do not commute. We
compare our field theory results with string theory states and find perfect
agreement.
