Crystal symmetry groups and their representations I lattice extensions of point groups in R2 and R3

We classify all point groups in R² and R³ according to their decomposition into m-fold semidirect products of cyclic groups. This product is characterized completely by means of m?1 matrices α⁽¹⁾,…, α^(m?1) with non-negative integer elements. For any arithmetic class Pⁿ_i] in Rⁿ (n=2, 3) we define the set D(Pⁿ_i, T) of representative translations with the following properties: (a) The elements of D(Pⁿ_i, T) are in one-to-one correspondence with defining relations of Pⁿ_i; (b) D(Pⁿ_i, T) determines uniquely all equivalence classes of the factors of Pⁿ_i over T.We present: (i) a construction of all sets D(Pⁿ_i, T) in R² and R³; (ii) a construction of all algebraic classes of space groups and a derivation of all group operations which transform one algebraic class into another; (iii) a derivation of formulas expressing the multiplication table of any space group by means of the representative translations and matrices α⁽¹⁾,…, α^(m?1). The results (i), (ii) and (iii) are obtained without making use of the notion of non-primitive translations.