  We study the bandgap structure of two-dimensional photonic crystals created
by a triangular lattice of rotated hexagonal holes, and explore the effects of
the reduced symmetry in the unit-cell geometry on the value of the absolute
bandgap and the frequencies of localized defect modes. We reveal that a maximum
absolute bandgap for this structure is achieved for an intermediate rotation
angle of the holes. This angle depends on the radius of the holes and the
refractive index of the background material. We also study the properties of
the defect modes created by missing holes, and discuss the mode tunability in
such structures.
