C6v E 2 C6 2 C3 C2 3 sv 3 sd <R> <p> <—d—> <——f——> <———g———> <————h————> <—————i—————> A1 1 1 1 1 1 1 ... ..T ....T ......T ........T ..........T T...........T A2 1 1 1 1 -1 -1 ..T ... ..... ....... ......... ........... .T........... B1 1 -1 1 -1 1 -1 ... ... ..... T...... ..T...... ....T...... ......T...... B2 1 -1 1 -1 -1 1 ... ... ..... .T..... ...T..... .....T..... .......T..... E1 2 1 -1 -2 0 0 TT. TT. ..TT. ....TT. ......TT. TT......TT. ..TT......TT. E2 2 -1 -1 2 0 0 ... ... TT... ..TT... TT..TT... ..TT..TT... ....TT..TT... Symmetry of Rotations and Cartesian products A1 p+d+f+g+h+2i+2j+2k+2l+2m z, z^{2}, z^{3}, z^{4}, z^{5}, x^{2}(x^{2}−3y^{2})^{2}−y^{2}(3x^{2}−y^{2})^{2}, z^{6} A2 R+i+j+k+l+m R_{z}, xy(x^{2}−3y^{2})(3x^{2}−y^{2}) B1 f+g+h+i+j+k+2l+2m x(x^{2}−3y^{2}), xz(x^{2}−3y^{2}), xz^{2}(x^{2}−3y^{2}), xz^{3}(x^{2}−3y^{2}) B2 f+g+h+i+j+k+2l+2m y(3x^{2}−y^{2}), yz(3x^{2}−y^{2}), yz^{2}(3x^{2}−y^{2}), yz^{3}(3x^{2}−y^{2}) E1 R+p+d+f+g+2h+2i+3j+3k+3l+3m {R_{x}, R_{y}}, {x, y}, {xz, yz}, {xz^{2}, yz^{2}}, {xz^{3}, yz^{3}}, {x(x^{2}−(5+2√5)y^{2})(x^{2}−(5−2√5)y^{2}), y((5+2√5)x^{2}−y^{2})((5−2√5)x^{2}−y^{2})}, {xz^{4}, yz^{4}}, {xz(x^{2}−(5+2√5)y^{2})(x^{2}−(5−2√5)y^{2}), yz((5+2√5)x^{2}−y^{2})((5−2√5)x^{2}−y^{2})}, {xz^{5}, yz^{5}} E2 d+f+2g+2h+2i+2j+3k+3l+4m {x^{2}−y^{2}, xy}, {z(x^{2}−y^{2}), xyz}, {(x^{2}−y^{2})^{2}−4x^{2}y^{2}, xy(x^{2}−y^{2})}, {z^{2}(x^{2}−y^{2}), xyz^{2}}, {z((x^{2}−y^{2})^{2}−4x^{2}y^{2}), xyz(x^{2}−y^{2})}, {z^{3}(x^{2}−y^{2}), xyz^{3}}, {z^{2}((x^{2}−y^{2})^{2}−4x^{2}y^{2}), xyz^{2}(x^{2}−y^{2})}, {z^{4}(x^{2}−y^{2}), xyz^{4}} Notes: α The order of the C_{6v} point group is 12, and the order of the principal axis (C_{6}) is 6. The group has 6 irreducible representations. β The C_{6v} point group is isomorphic to D_{3d}, D_{3h} and D_{6}. γ The C_{6v} point group is generated by two symmetry elements, C_{6} and any σ_{v} (or, non-canonically, any σ_{d}). Also, the group may be generated from a σ_{v} plus a σ_{d} (some pairs will yield smaller groups, though; choosing a minimum angle is safe). δ There are two different sets of symmetry planes containing the principal axis (z axis in standard orientation). By convention, the set denoted as σ_{v} has the xz plane as a member, while the yz plane is a member of the σ_{d} set. ε The lowest nonvanishing multipole moment in C_{6v} is 2 (dipole moment). ζ This point group is non-Abelian (some symmetry operations are not commutative). Therefore, the character table contains multi-membered classes and degenerate irreducible representations. η All characters are integers because the order of the principal axis is 1,2,3,4 or 6. This implies that the point group corresponds to a constructible polygon which can be used for tiling the plane. Such point groups are also referred to as “crystallographic point groups”, as they are compatible with periodic lattice symmetry. There are exactly 32 such groups: C_{1},C_{s},C_{i},C_{2},C_{2h},C_{2v},C_{3},C_{3h},C_{3v},C_{4},C_{4h},C_{4v},C_{6},C_{6h},C_{6v},D_{2},D_{2d},D_{2h},D_{3},D_{3d},D_{3h},D_{4},D_{4h},D_{6},D_{6h},S_{4},S_{6},T,T_{d},T_{h},O,O_{h}.
C_{4v} | ||
C_{5v} | ||
C_{6} | C_{6v} | C_{6h} D_{6} D_{6h} D_{6d} S_{6} |
C_{7v} | ||
C_{8v} |
This Character Table for the C_{6v} point group was created by Gernot Katzer.
For other groups and some explanations, see the Main Page.