D11 E 2 C11 2 C11^2 2 C11^3 2 C11^4 2 C11^5 11 C2' <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 A2 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 -1.0000 ..T ..T ..... ......T ......... ..........T ............. E1 2.0000 1.6825 0.8308 -0.2846 -1.3097 -1.9189 0.0000 TT. TT. ..TT. ....TT. ......TT. ........TT. ..........TT. E2 2.0000 0.8308 -1.3097 -1.9189 -0.2846 1.6825 0.0000 ... ... TT... ..TT... ....TT... ......TT... ........TT... E3 2.0000 -0.2846 -1.9189 0.8308 1.6825 -1.3097 0.0000 ... ... ..... TT..... ..TT..... ....TT..... ......TT..... E4 2.0000 -1.3097 -0.2846 1.6825 -1.9189 0.8308 0.0000 ... ... ..... ....... TT....... ..TT....... ....TT....... E5 2.0000 -1.9189 1.6825 -1.3097 0.8308 -0.2846 0.0000 ... ... ..... ....... ......... TT......... TTTT......... Irrational character values: 1.682507065662 = 2*cos(2*π/11) 0.830830026004 = 2*cos(4*π/11) -0.284629676547 = 2*cos(6*π/11) -1.309721467891 = 2*cos(8*π/11) -1.918985947229 = 2*cos(10*π/11) Symmetry of Rotations and Cartesian products A1 d+g+i+k+m z^{2}, z^{4}, z^{6} A2 R+p+f+h+j+l R_{z}, z, z^{3}, z^{5} E1 R+p+d+f+g+h+i+j+k+l+2m {R_{x}, R_{y}}, {x, y}, {xz, yz}, {xz^{2}, yz^{2}}, {xz^{3}, yz^{3}}, {xz^{4}, yz^{4}}, {xz^{5}, yz^{5}} E2 d+f+g+h+i+j+k+2l+2m {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}}, {z^{4}(x^{2}−y^{2}), xyz^{4}} E3 f+g+h+i+j+2k+2l+2m {x(x^{2}−3y^{2}), y(3x^{2}−y^{2})}, {xz(x^{2}−3y^{2}), yz(3x^{2}−y^{2})}, {xz^{2}(x^{2}−3y^{2}), yz^{2}(3x^{2}−y^{2})}, {xz^{3}(x^{2}−3y^{2}), yz^{3}(3x^{2}−y^{2})} E4 g+h+i+2j+2k+2l+2m {(x^{2}−y^{2})^{2}−4x^{2}y^{2}, xy(x^{2}−y^{2})}, {z((x^{2}−y^{2})^{2}−4x^{2}y^{2}), xyz(x^{2}−y^{2})}, {z^{2}((x^{2}−y^{2})^{2}−4x^{2}y^{2}), xyz^{2}(x^{2}−y^{2})} E5 h+2i+2j+2k+2l+2m {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})}, {x^{2}(x^{2}−3y^{2})^{2}−y^{2}(3x^{2}−y^{2})^{2}, xy(x^{2}−3y^{2})(3x^{2}−y^{2})}, {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})} Notes: α The order of the D_{11} point group is 22, and the order of the principal axis (C_{11}) is 11. The group has 7 irreducible representations. β The D_{11} point group is isomorphic to C_{11v}. γ The D_{11} point group is generated by two symmetry elements, C_{11} and a perpendicular C_{2}^{′}. Also, the group may be generated from any two C_{2}^{′} axes. δ The group contains one set of C_{2}^{′} symmetry axes perpendicular to the principal (z) axis. The x axis (but not the y axis) is a member of that set. ε The lowest nonvanishing multipole moment in D_{11} 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 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 just less than half the order of the principal axis. For this group, however, none of the irrational characters can be expressed by a closed algebraic form using real numbers only. ι The regular hendecagon is the smallest regular polygon not constructible with ruler, compass and angle trisector. This is because 2*cos(2*π/11) has algebraic degree of five, being the solution of an irreducible quintic equation. Because this quintic equation is solvable, the value of cos(2*π/11) can be expressed using square and fifth roots and complex numbers. That algebraic form is, however, very complex and thus not shown here.
D_{9} | ||
D_{10} | ||
C_{11} C_{11v} C_{11h} | D_{11} | D_{11h} D_{11d} |
D_{12} | ||
D_{13} |
This Character Table for the D_{11} point group was created by Gernot Katzer.
For other groups and some explanations, see the Main Page.