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 [\geq] 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 = 4N])
n [C_{n}] n n-gonal pyramid Regular n-gon 4
[\overline{n}] [S_{n}] n [{1 \over 2}n]-gonal streptohedron [{1 \over 2}n]-gonal antiprism [\overline{4}]
[n/m] [C_{nh}] 2n n-gonal dipyramid n-gonal prism [4/m]
n22 [D_{n}] 2n n-gonal trapezohedron Twisted n-gonal antiprism 422
nmm [C_{nv}] 2n Di-n-gonal pyramid Truncated n-gon 4mm
[\overline{n}2m] [D_{{1 \over 2}nd}] 2n n-gonal scalenohedron [{1 \over 2}n]-gonal antiprism sliced off by pinacoid [\overline{4}2m]
[n/mmm, \ \displaystyle{n \over m}{2 \over m}{2 \over m}] [D_{nh}] 4n Di-n-gonal dipyramid Edge-truncated n-gonal prism [4/mmm]
[(2N + 1)]-gonal system (single n-fold symmetry axis with [n = 2N + 1])
n [C_{n}] n n-gonal pyramid Regular n-gon 1, 3
[\overline{n} = n \times \overline{1}] [C_{ni}] 2n n-gonal streptohedron n-gonal antiprism [\overline{1},\ \overline{3} = 3 \times \overline{1}]
n2 [D_{n}] 2n n-gonal trapezohedron Twisted n-gonal antiprism 32
nm [C_{nv}] 2n Di-n-gonal pyramid Truncated n-gon 3m
[\overline{n}m, \ \overline{n} \displaystyle{2 \over m}] [D_{nd}] 4n Di-n-gonal scalenohedron n-gonal antiprism sliced off by pinacoid [\overline{3}m]
[(4N + 2)]-gonal system (single n-fold symmetry axis with [n = 4N + 2])
n [C_{n}] n n-gonal pyramid Regular n-gon 2, 6
[\overline{n} = {\textstyle{1 \over 2}}n/m] [C_{{1 \over 2}nh}] n [{\textstyle{1 \over 2}}n]-gonal dipyramid [{\textstyle{1 \over 2}}n]-gonal prism [\overline2 \equiv m, \ \overline{6} \equiv 3/m]
[n/m] [C_{nh}] 2n n-gonal dipyramid n-gonal prism [2/m, \ 6/m]
n22 [D_{n}] 2n n-gonal trapezohedron Twisted n-gonal antiprism 222, 622
nmm [C_{nv}] 2n Di-n-gonal pyramid Truncated n-gon mm2, 6mm
[\overline{n}2m = {\textstyle{1 \over 2}}n/m2m] [D_{{1 \over 2}nh}] 2n Di-[{\textstyle{1 \over 2}}n]-gonal dipyramid Truncated [{\textstyle{1 \over 2}}n]-gonal prism [\overline{6}2m]
[n/mmm, \ \displaystyle{n \over m}{2 \over m}{2 \over m}] [D_{nh}] 4n Di-n-gonal dipyramid Edge-truncated n-gonal prism mmm, [6/mmm]
Cubic system (for details see Table 3.2.3.2[link])
23 T 12 Pentagon-tritetrahedron Snub tetrahedron 23
[m\overline{3}, \displaystyle{2 \over m}\overline{3}] [T_{h}] 24 Didodecahedron Cube & octahedron & pentagon-dodecahedron [m\overline{3}]
432 O 24 Pentagon-trioctahedron Snub cube 432
[\overline{4}3m] [T_{d}] 24 Hexatetrahedron Cube truncated by two tetrahedra [\overline{4}3m]
[m\overline{3}m, \ \displaystyle{4 \over m}\overline{3}{2 \over m}] [O_{h}] 48 Hexaoctahedron Cube truncated by octahedron and by rhomb-dodecahedron [m\overline{3}m]
Icosahedral system (for details see Table 3.2.3.3[link])
235 I 60 Pentagon-hexecontahedron Snub pentagon-dodecahedron
[m\overline{3}\overline{5}, \ {\displaystyle{2 \over m}}\overline{3}\overline{5}] [I_{h}] 120 Hecatonicosahedron Pentagon-dodecahedron truncated by icosahedron and by rhomb-triacontahedron
Cylindrical system
[C_{\infty}] Rotating cone Rotating circle
[\infty/m \equiv \overline{\infty}] [C_{\infty h} \equiv S_{\infty} \equiv C_{\infty i}] Rotating double cone Rotating finite cylinder
∞2 [D_{\infty}] `Anti-rotating' double cone `Anti-rotating' finite cylinder
m [C_{\infty v}] Stationary cone Stationary circle
[\infty/mm \equiv \overline{\infty}m, \ {\displaystyle{\infty \over m}{2 \over m}} \equiv \overline{\infty} {\displaystyle{2 \over m}}] [D_{\infty h} \equiv D_{\infty d}] Stationary double cone Stationary finite cylinder
Spherical system§
[2 \infty], [\infty\infty] K Rotating sphere Rotating sphere
[m \overline{\infty},\ {\displaystyle{2 \over m}} \overline{\infty},\ \infty\infty m] [K_{h}] Stationary sphere Stationary sphere
The Hermann–Mauguin symbols of the two icosahedral point groups are often written as 532 and [\bar{5}\bar{3}m] (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 [\infty 2], enantiomorphism occurs, i.e. forms with opposite senses of rotation. Class [\infty/m \equiv \overline{\infty}] 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 [\infty 2] 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 [2 \infty] 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[link], Optical properties). The stationary spheres in class [m \overline{\infty}] contain infinitely many mirror planes through the centres of the spheres. Group [2 \infty] is sometimes symbolized by [\infty \infty]; group [m \overline{\infty}] by [\overline{\infty}\, \overline{\infty}] or [\infty \infty m]. 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.