International
Tables for Crystallography Volume A Spacegroup symmetry Edited by Th. Hahn © International Union of Crystallography 2006 
International Tables for Crystallography (2006). Vol. A, ch. 15.4, pp. 904905
https://doi.org/10.1107/97809553602060000536 Chapter 15.4. Normalizers of point groups^{a}Institut für Mineralogie, Petrologie und Kristallographie, PhilippsUniversität, D35032 Marburg, Germany In Chapter 15.4, the normalizers of the twodimensional point groups with respect to the full isometry group of the circle and of the threedimensional point groups with respect to the full isometry group of the sphere are tabulated. 
Normalizers with respect to the Euclidean or affine group may be defined for any group of isometries (cf. Gubler, 1982a,b). For a point group, however, it seems inadequate to use a supergroup that contains transformations that do not map a fixed point of that point group onto itself. Appropriate supergroups for the definition of normalizers of point groups are the full isometry groups of the sphere, , and of the circle, ∞m, in threedimensional and twodimensional space (cf. Galiulin, 1978).
These normalizers are listed in Tables 15.4.1.1 and 15.4.1.2. It has to be noticed that the normalizer of a crystallographic point group may contain continuous rotations, i.e. rotations with infinitesimal rotation angle, or noncrystallographic rotations (; ; ). In analogy to space groups, these normalizers define equivalence relationships on the `Wyckoff positions' of the point groups (cf. Section 10.1.2 ). They give also the relation between the different but equivalent morphological descriptions of a crystal.


References
Galiulin, R. V. (1978). Holohedral varieties of simple forms of crystals. Sov. Phys. Crystallogr. 23, 635–641.Gubler, M. (1982a). Über die Symmetrien der Symmetriegruppen: Automorphismengruppen, Normalisatorgruppen und charakteristische Untergruppen von Symmetriegruppen, insbesondere der kristallographischen Punkt und Raumgruppen. Dissertation, University of Zürich, Switzerland.
Gubler, M. (1982b). Normalizer groups and automorphism groups of symmetry groups. Z. Kristallogr. 158, 1–26.