D1h E C2' sh sv <R> <p> <—d—> <——f——> <———g———> <————h————> <—————i—————> A1' 1 1 1 1 ... T.. T...T T...T.. T...T...T T...T...T.. T...T...T...T A1" 1 1 -1 -1 T.. ... ...T. ...T... ...T...T. ...T...T... ...T...T...T. A2' 1 -1 1 -1 ..T .T. .T... .T...T. .T...T... .T...T...T. .T...T...T... A2" 1 -1 -1 1 .T. ..T ..T.. ..T...T ..T...T.. ..T...T...T ..T...T...T.. Symmetry of Rotations and Cartesian products A1' p+2d+2f+3g+3h+4i+4j+5k+5l+6m x, x^{2}−y^{2}, z^{2}, x(x^{2}−3y^{2}), xz^{2}, (x^{2}−y^{2})^{2}−4x^{2}y^{2}, z^{2}(x^{2}−y^{2}), z^{4}, x(x^{2}−(5+2√5)y^{2})(x^{2}−(5−2√5)y^{2}), xz^{2}(x^{2}−3y^{2}), xz^{4}, x^{2}(x^{2}−3y^{2})^{2}−y^{2}(3x^{2}−y^{2})^{2}, z^{2}((x^{2}−y^{2})^{2}−4x^{2}y^{2}), z^{4}(x^{2}−y^{2}), z^{6} A1" R+d+f+2g+2h+3i+3j+4k+4l+5m R_{x}, yz, xyz, yz(3x^{2}−y^{2}), yz^{3}, xyz(x^{2}−y^{2}), xyz^{3}, yz((5+2√5)x^{2}−y^{2})((5−2√5)x^{2}−y^{2}), yz^{3}(3x^{2}−y^{2}), yz^{5} A2' R+p+d+2f+2g+3h+3i+4j+4k+5l+5m R_{z}, y, xy, y(3x^{2}−y^{2}), yz^{2}, xy(x^{2}−y^{2}), xyz^{2}, y((5+2√5)x^{2}−y^{2})((5−2√5)x^{2}−y^{2}), yz^{2}(3x^{2}−y^{2}), yz^{4}, xy(x^{2}−3y^{2})(3x^{2}−y^{2}), xyz^{2}(x^{2}−y^{2}), xyz^{4} A2" R+p+d+2f+2g+3h+3i+4j+4k+5l+5m R_{y}, z, xz, z(x^{2}−y^{2}), z^{3}, xz(x^{2}−3y^{2}), xz^{3}, z((x^{2}−y^{2})^{2}−4x^{2}y^{2}), z^{3}(x^{2}−y^{2}), z^{5}, xz(x^{2}−(5+2√5)y^{2})(x^{2}−(5−2√5)y^{2}), xz^{3}(x^{2}−3y^{2}), xz^{5} Notes: α The order of the D_{1h} point group is 4, and the order of the principal axis (C_{2}^{′}) is 2. The group has 4 irreducible representations. β The D_{1h} point group is identical to C_{2v} in non-standard orientation (the C_{2}^{′} axis is x, the σ_{h} plane is xy and the σ_{v} plane is xz). γ The lowest nonvanishing multipole moment in D_{1h} is 2 (dipole moment). δ This is an Abelian point group (the commutative law holds between all symmetry operations). The D_{1h} group is Abelian because it satisfies the sufficient condition to contain no axes of order higher than two. In Abelian groups, all symmetry operations form a class of their own, and all irreducible representations are one-dimensional. ε There are no symmetry elements of an order higher than 2 in this group. The symmetry-adapted Cartesian products in the table above are needlessly complicated; rather, any simple product will do. ζ All characters are integers because the order of the principal axis is 1,2,3,4 or 6. 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_{1} C_{1v} C_{s} D_{1} | D_{1h} | D_{1d} |
D_{2h} | ||
D_{3h} |
This Character Table for the D_{1h} point group was created by Gernot Katzer.
For other groups and some explanations, see the Main Page.