International
Tables for
Crystallography
Volume A
Space-group symmetry
Edited by M. I. Aroyo

International Tables for Crystallography (2016). Vol. A, ch. 3.2, p. 734

Table 3.2.1.6

Th. Hahna and H. Klappera
 Table 3.2.1.6| top | pdf | Classes of general point groups in three dimensions (N = integer 0)
Short general Hermann–Mauguin symbol, followed by full symbol where differentSchoenflies symbolOrder of groupGeneral face formGeneral point formCrystallographic groups
4N-gonal system (single n-fold symmetry axis with )
n n n-gonal pyramid Regular n-gon 4
n -gonal streptohedron -gonal antiprism
2n n-gonal dipyramid n-gonal prism
n22 2n n-gonal trapezohedron Twisted n-gonal antiprism 422
nmm 2n Di-n-gonal pyramid Truncated n-gon 4mm
2n n-gonal scalenohedron -gonal antiprism sliced off by pinacoid
4n Di-n-gonal dipyramid Edge-truncated n-gonal prism
-gonal system (single n-fold symmetry axis with )
n n n-gonal pyramid Regular n-gon 1, 3
2n n-gonal streptohedron n-gonal antiprism
n2 2n n-gonal trapezohedron Twisted n-gonal antiprism 32
nm 2n Di-n-gonal pyramid Truncated n-gon 3m
4n Di-n-gonal scalenohedron n-gonal antiprism sliced off by pinacoid
-gonal system (single n-fold symmetry axis with )
n n n-gonal pyramid Regular n-gon 2, 6
n -gonal dipyramid -gonal prism
2n n-gonal dipyramid n-gonal prism
n22 2n n-gonal trapezohedron Twisted n-gonal antiprism 222, 622
nmm 2n Di-n-gonal pyramid Truncated n-gon mm2, 6mm
2n Di--gonal dipyramid Truncated -gonal prism
4n Di-n-gonal dipyramid Edge-truncated n-gonal prism mmm,
Cubic system (for details see Table 3.2.3.2)
23 T 12 Pentagon-tritetrahedron Snub tetrahedron 23
24 Didodecahedron Cube & octahedron & pentagon-dodecahedron
432 O 24 Pentagon-trioctahedron Snub cube 432
24 Hexatetrahedron Cube truncated by two tetrahedra
48 Hexaoctahedron Cube truncated by octahedron and by rhomb-dodecahedron
Icosahedral system (for details see Table 3.2.3.3)
235 I 60 Pentagon-hexecontahedron Snub pentagon-dodecahedron
120 Hecatonicosahedron Pentagon-dodecahedron truncated by icosahedron and by rhomb-triacontahedron
Cylindrical system
Rotating cone Rotating circle
Rotating double cone Rotating finite cylinder
∞2 Anti-rotating' double cone Anti-rotating' finite cylinder
m Stationary cone Stationary circle
Stationary double cone Stationary finite cylinder
Spherical system§
, K Rotating sphere Rotating sphere
Stationary sphere Stationary sphere
The Hermann–Mauguin symbols of the two icosahedral point groups are often written as 532 and (see text).
Rotating and anti-rotating' forms in the cylindrical system have no vertical' mirror planes, whereas stationary forms have infinitely many vertical mirror planes. In classes ∞ and , enantiomorphism occurs, i.e. forms with opposite senses of rotation. Class exhibits no enantiomorphism due to the centre of symmetry, even though the double cone is rotating in one direction. This can be understood as follows: The handedness of a rotating cone depends on the sense of rotation with respect to the axial direction from the base to the tip of the cone. Thus, the rotating double cone consists of two cones with opposite handedness and opposite orientations related by the (single) horizontal mirror plane. In contrast, the anti-rotating' double cone in class consists of two cones of equal handedness and opposite orientations, which are related by the (infinitely many) twofold axes. The term anti-rotating' means that upper and lower halves of the forms rotate in opposite directions.
§The spheres in class of the spherical system must rotate around an axis with at least two different orientations, in order to suppress all mirror planes. This class exhibits enantiomorphism, i.e. it contains spheres with either right-handed or left-handed senses of rotation around the axes (cf. Section 3.2.2.4, Optical properties). The stationary spheres in class contain infinitely many mirror planes through the centres of the spheres. Group is sometimes symbolized by ; group by or . The symbols used here indicate the minimal symmetry necessary to generate the groups; they show, furthermore, the relation to the cubic groups. The Schoenflies symbol K is derived from the German name Kugelgruppe.