The Graph of a Geometric Class of Space Groups

Some comments on the subgroup relations of all space groups within a geometric class are given. A graph is defined that encodes the essential inclusions of these groups in one another. In a three-dimensional space, such a construction answers, amongst others, the question when one space group occurs as a maximal subgroup of another in the same geometric class. This information can also be obtained in higher dimensions in CARAT, which provides an environment for constructing, counting, and recognizing space groups up to dimension six.