vector field has this symmetry group. the group generated by all translations and reflections in points; they are isomorphic with the generalized dihedral group Dih(R). Two dimensions Up to conjugacy the discrete point groups in two-dimensional space are the following classes: cyclic groups C1, C2, C3, C4, ... where Cn consists of all rotations about a fixed point by multiples of the angle 360°/n dihedral groups D1, D2, D3, D4, ..., where Dn (of order 2n) consists of the rotations in Cn together with reflections in n axes that pass through the fixed point. C1 is the trivial group containing only the identity operation, which occurs when the figure is asymmetric, for example the letter "F". C2 is the symmetry group of the letter "Z", C3 that of a triskelion, C4 of a swastika, and C5, C6, etc. are the symmetry groups of similar swastika-like figures with five, six, etc. arms instead of four. D1 is the 2-element group containing the identity operation and a single reflection, which occurs when the figure has only a single axis of bilateral symmetry, for example the letter "A". D2, which is isomorphic to the Klein four-group, is the symmetry group of a non-equilateral rectangle. This figure has four symmetry operations: the identity operation, one twofold axis of rotation, and two nonequivalent mirror planes. D3, D4 etc. are the symmetry groups of the regular polygons. Within each of these symmetry types, there are two degrees of freedom for the center of rotation, and in the case of the dihedral groups, one more for the positions of the mirrors. The remaining isometry groups in two dimensions with a fixed point are: the special orthogonal group SO(2) consisting of all rotations about a fixed point; it is also called the circle group S1, the multiplicative group of complex numbers of absolute value 1. It is the proper symmetry group of a circle and the continuous equivalent of Cn. There is no geometric figure that has as full symmetry group the circle group, but for a vector field it may apply (see the three-dimensional case below). the orthogonal group O(2) consisting of all rotations about a fixed point and reflections in any axis through that fixed point. This is the symmetry group of a circle. It is also called Dih(S1) as it is the generalized dihedral group of S1. Non-bounded figures may have isometry groups including translations;