D4d E 2 S8 2 C4 2 S8^3 C2 4 C2' 4 sd <R> <p> <—d—> <——f——> <———g———> <————h————> <—————i—————> A1 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 ... ... ....T ....... ........T ...T....... ............T A2 1.0000 1.0000 1.0000 1.0000 1.0000 -1.0000 -1.0000 ..T ... ..... ....... ......... ..T........ ............. B1 1.0000 -1.0000 1.0000 -1.0000 1.0000 1.0000 -1.0000 ... ... ..... ....... T........ ........... ....T........ B2 1.0000 -1.0000 1.0000 -1.0000 1.0000 -1.0000 1.0000 ... ..T ..... ......T .T....... ..........T .....T....... E1 2.0000 1.4142 0.0000 -1.4142 -2.0000 0.0000 0.0000 ... TT. ..... ....TT. ..TT..... ........TT. ..TT..TT..... E2 2.0000 0.0000 -2.0000 0.0000 2.0000 0.0000 0.0000 ... ... TT... ..TT... ....TT... ......TT... TT......TT... E3 2.0000 -1.4142 0.0000 1.4142 -2.0000 0.0000 0.0000 TT. ... ..TT. TT..... ......TT. TT..TT..... ..........TT. Irrational character values: 1.414213562373 = 2*cos(2*π/8) = 2*cos(π/4) = √2 Symmetry of Rotations and Cartesian products A1 d+g+h+i+j+2k+l+2m z^{2}, z^{4}, xyz(x^{2}−y^{2}), z^{6} A2 R+h+j+k+l+m R_{z}, z((x^{2}−y^{2})^{2}−4x^{2}y^{2}) B1 g+i+k+l+m (x^{2}−y^{2})^{2}−4x^{2}y^{2}, z^{2}((x^{2}−y^{2})^{2}−4x^{2}y^{2}) B2 p+f+g+h+i+j+k+2l+m z, z^{3}, xy(x^{2}−y^{2}), z^{5}, xyz^{2}(x^{2}−y^{2}) E1 p+f+g+h+2i+2j+2k+3l+2m {x, y}, {xz^{2}, yz^{2}}, {xz(x^{2}−3y^{2}), yz(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})} E2 d+f+g+h+2i+2j+2k+2l+3m {x^{2}−y^{2}, xy}, {z(x^{2}−y^{2}), xyz}, {z^{2}(x^{2}−y^{2}), xyz^{2}}, {z^{3}(x^{2}−y^{2}), xyz^{3}}, {x^{2}(x^{2}−3y^{2})^{2}−y^{2}(3x^{2}−y^{2})^{2}, xy(x^{2}−3y^{2})(3x^{2}−y^{2})}, {z^{4}(x^{2}−y^{2}), xyz^{4}} E3 R+d+f+g+2h+i+2j+2k+2l+3m {R_{x}, R_{y}}, {xz, yz}, {x(x^{2}−3y^{2}), y(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^{5}, yz^{5}} Notes: α The order of the D_{4d} point group is 16, and the order of the principal axis (S_{8}) is 8. The group has 7 irreducible representations. β The D_{4d} point group is isomorphic to C_{8v} and D_{8}. γ The D_{4d} point group is generated by two symmetry elements, S_{8} and either a perpendicular C_{2}^{′} or a vertical σ_{d}. Also, the group may be generated from any C_{2}^{′} plus any σ_{d} plane. δ The group contains one set of twofold symmetry axes (C_{2}^{′}) perpendicular to the principal (z) axis. Both x and y axes are members of that set. ε The single σ_{d} set of symmetry planes contains neither the xz nor the yz planes. ζ The lowest nonvanishing multipole moment in D_{4d} is 4 (quadrupole moment). η This point group is non-Abelian (some symmetry operations are not commutative). Therefore, the character table contains multi-membered classes and degenerate irreducible representations. θ Some of the characters in the table are irrational because the order of the principal axis is neither 1,2,3,4 nor 6. These irrational values can be expressed as cosine values, or as solutions of algebraic equations with a leading coefficient of 1. All characters are algebraic integers of a degree much less than half the order of the principal axis. ι The point group corresponds to a constructible polygon, as the order of the principal axis is a product of any number of different Fermat primes (3,5,17,257,65537) times an arbitrary power of two. Therefore, all characters have an algebraic degree which is a power of two and can be expressed as radicals involving only square roots and integer numbers.
D_{2d} | ||
D_{3d} | ||
C_{4} C_{4v} C_{4h} D_{4} D_{4h} | D_{4d} | S_{4} |
D_{5d} | ||
D_{6d} |
This Character Table for the D_{4d} point group was created by Gernot Katzer.
For other groups and some explanations, see the Main Page.