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Crystals which have a uniform distribution of defects are endowed with a Liegroup description which allows one to construct an associated discretestructure. These structures are in fact the discrete subgroups of the ambientLie group. The geometrical symmetries of these structures can be computed interms of the changes of generators of the discrete subgroup which preserve thediscrete set of points. Here a classification of the symmetries for thediscrete subgroups of a particular class of three-dimensional solvable Liegroup is presented. It is a fact that there are only three mathematicallydistinct types of Lie groups which model uniform defective crystals, and thecalculations given here complete the discussion of the symmetries of thecorresponding discrete structures. We show that those symmetries correspondingto automorphisms of the discrete subgroups extend uniquely to symmetries of theambient Lie group and we regard these symmetries as (restrictions of) elasticdeformations of the continuous defective crystal. Other symmetries of thediscrete structures are classified as `inelastic' symmetries.

A classification of the symmetries of uniform discrete defective crystals