Weak Cayley Table Groups of Crystallographic Groups I

Let Formula: see text be a group. A weak Cayley table isomorphism is a bijection Formula: see text such that (i) Formula: see text is conjugate to Formula: see text for Formula: see text, Formula: see text and (ii) Formula: see text sends conjugacy classes to conjugacy classes. The set of all such bijections forms a group Formula: see text. We study Formula: see text for 56 of the 219 three-dimensional crystallographic groups, as well as some other groups. These 56 groups are related to our previous work on wallpaper groups. In this paper we determine for which of the 56 groups Formula: see text has elements that are not automorphisms nor anti-automorphisms.