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In three dimensional geometry, there are four infinite series of point groups in three dimensions (n>=1) with n-fold rotational or reflectional symmetry about one axis (by an angle of 360°/n) that does not change the object.
Cnv, [n], (*nn) of order 2n - pyramidal symmetry or full acro-n-gonal group (abstract group Dihn); in biology C2v is called biradial symmetry. For n=1 we have again Cs (1*). It has vertical mirror planes. This is the symmetry group for a regular n-sided pyramid.
S2n, [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 Z2n); 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 Dnd, it contains a number of improper rotations without containing the corresponding rotations.
C2h, [2,2+] (2*) and C2v, , (*22) of order 4 are two of the three 3D symmetry group types with the Klein four-group as abstract group. C2v applies e.g. for a rectangular tile with its top side different from its bottom side.
In the limit these four groups represent Euclidean plane frieze groups as C?, C?h, C?v, and S?. Rotations become translations in the limit. Portions of the infinite plane can also be cut and connected into an infinite cylinder.