Cyclic symmetry in three dimensions

They are the finite symmetry groups on a cone. For n = ∞ they correspond to four frieze groups. Schönflies notation is used. The terms horizontal (h) and vertical (v) imply the existence and direction of reflections with respect to a vertical axis of symmetry. Also shown are Coxeter notation in brackets, and, in parentheses, orbifold notation.

Types

Chiral

• C_n, [n]⁺, (nn) of order n - n-fold rotational symmetry - acro-n-gonal group (abstract group Z_n); for n=1: no symmetry (trivial group)

Achiral

• C_nh, [n⁺,2], (n*) of order 2n - prismatic symmetry or ortho-n-gonal group (abstract group Z_n × Dih₁); for n=1 this is denoted by C_s (1*) and called reflection symmetry, also bilateral symmetry. It has reflection symmetry with respect to a plane perpendicular to the n-fold rotation axis.
• C_nv, [n], (*nn) of order 2n - pyramidal symmetry or full acro-n-gonal group (abstract group Dih_n); in biology C_2v is called biradial symmetry. For n=1 we have again C_s (1*). It has vertical mirror planes. This is the symmetry group for a regular n-sided pyramid.
• S_2n, [2⁺,2n⁺], (n×) of order 2n - gyro-n-gonal group (not to be confused with symmetric groups, for which the same notation is used; abstract group Z_2n); It has a 2n-fold rotoreflection axis, also called 2n-fold improper rotation axis, i.e., the symmetry group contains a combination of a reflection in the horizontal plane and a rotation by an angle 180°/n. Thus, like D_nd, it contains a number of improper rotations without containing the corresponding rotations.
  • for n=1 we have S₂ (1×), also denoted by C_i; this is inversion symmetry.

C_2h, [2,2⁺] (2*) and C_2v, [2], (*22) of order 4 are two of the three 3D symmetry group types with the Klein four-group as abstract group. C_2v applies e.g. for a rectangular tile with its top side different from its bottom side.