  We combine numerical and analytical methods to study two dimensional active
crystals formed by permanently linked swimmers and with two distinct alignment
interactions. The system admits a stationary phase with quasi long range
translational order, as well as a moving phase with quasi-long range velocity
order. The translational order in the moving phase is significantly influenced
by alignment interaction. For Vicsek-like alignment, the translational order is
short-ranged, whereas the bond-orientational order is quasi-long ranged,
implying a moving hexatic phase. For elasticity-based alignment, the
translational order is quasi-long ranged parallel to the motion and
short-ranged in perpendicular direction, whereas the bond orientational order
is long-ranged. We also generalize these results to higher dimensions.
