On equivalent subgroups and supergroups of the space groups

Abstract

Symmetry operators (α\τα) of a space group G and (β\τβ) of a subgroup g are conjugated with respect to a similarity operator S = (S|T) where S is a 3 x 3 matrix and T a column matrix. The matrix S relates the lattice vectors of G to those of g whilst T describes the origin of the lattice of g in the coordinate system of G. We determine here the coefficients of the matrix S when G and g are equivalent, i.e. isosymbolic or enantiomorphic. Here the index of g in G is equal to |det S|. Its lowest values are tabulated for all space groups.