We present a complete formulation of the two-dimensional and three-dimensional crystallographic space groups in the conformal geometric algebra of Euclidean space. This enables a simple new representation of translational and orthogonal symmetries in a multiplicative group of versors. The generators of each group are constructed directly from a basis of lattice vectors that define its crystal class. A new system of space group symbols enables one to unambiguously write down all generators of a given space group directly from its symbol.
Most of the current work on conformal GA can be found at the website in Ref. 3 and links therein.