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.