D4 E 2 C4 C2 2 C2' 2 C2" <R> <p> <—d—> <——f——> <———g———> <————h————> <—————i—————> A1 1 1 1 1 1 ... ... ....T ....... T.......T ...T....... ....T.......T A2 1 1 1 -1 -1 ..T ..T ..... ......T .T....... ..T.......T .....T....... B1 1 -1 1 1 -1 ... ... T.... ...T... ....T.... .......T... T.......T.... B2 1 -1 1 -1 1 ... ... .T... ..T.... .....T... ......T.... .T.......T... E 2 0 -2 0 0 TT. TT. ..TT. TT..TT. ..TT..TT. TT..TT..TT. ..TT..TT..TT. Symmetry of Rotations and Cartesian products A1 d+2g+h+2i+j+3k+2l+3m z^{2}, (x^{2}−y^{2})^{2}−4x^{2}y^{2}, z^{4}, xyz(x^{2}−y^{2}), z^{2}((x^{2}−y^{2})^{2}−4x^{2}y^{2}), z^{6} A2 R+p+f+g+2h+i+2j+2k+3l+2m R_{z}, z, z^{3}, xy(x^{2}−y^{2}), z((x^{2}−y^{2})^{2}−4x^{2}y^{2}), z^{5}, xyz^{2}(x^{2}−y^{2}) B1 d+f+g+h+2i+2j+2k+2l+3m x^{2}−y^{2}, xyz, z^{2}(x^{2}−y^{2}), xyz^{3}, x^{2}(x^{2}−3y^{2})^{2}−y^{2}(3x^{2}−y^{2})^{2}, z^{4}(x^{2}−y^{2}) B2 d+f+g+h+2i+2j+2k+2l+3m xy, z(x^{2}−y^{2}), xyz^{2}, z^{3}(x^{2}−y^{2}), xy(x^{2}−3y^{2})(3x^{2}−y^{2}), xyz^{4} E R+p+d+2f+2g+3h+3i+4j+4k+5l+5m {R_{x}, R_{y}}, {x, y}, {xz, yz}, {x(x^{2}−3y^{2}), y(3x^{2}−y^{2})}, {xz^{2}, yz^{2}}, {xz(x^{2}−3y^{2}), yz(3x^{2}−y^{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^{2}(x^{2}−3y^{2}), yz^{2}(3x^{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^{3}(x^{2}−3y^{2}), yz^{3}(3x^{2}−y^{2})}, {xz^{5}, yz^{5}} Notes: α The order of the D_{4} point group is 8, and the order of the principal axis (C_{4}) is 4. The group has 5 irreducible representations. β The D_{4} point group is isomorphic to D_{2d} and C_{4v}. γ The D_{4} point group is generated by two symmetry elements, C_{4} and a perpendicular C_{2}^{′} (or, non-canonically, C_{2}^{″}). Also, the group may be generated from any C_{2}^{′} plus any C_{2}^{″} axes. δ There are two different sets of twofold symmetry axes perpendicular to the principal axis (z axis in standard orientation). By convention, the set denoted as C_{2}^{′} contains both the x and y axes. ε The lowest nonvanishing multipole moment in D_{4} is 4 (quadrupole 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 The point group is chiral, as it does not contain any mirroring operation. θ 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}.
D_{2} | ||
D_{3} | ||
C_{4} C_{4v} C_{4h} | D_{4} | D_{4h} D_{4d} S_{4} |
D_{5} | ||
D_{6} |
This Character Table for the D_{4} point group was created by Gernot Katzer.
For other groups and some explanations, see the Main Page.