C9v E 2 C9 2 C9^2 2 C3 2 C9^4 9 sv <R> <p> <—d—> <——f——> <———g———> <————h————> <—————i—————> A1 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 ... ..T ....T ......T ........T ..........T ............T A2 1.0000 1.0000 1.0000 1.0000 1.0000 -1.0000 ..T ... ..... ....... ......... ........... ............. E1 2.0000 1.5320 0.3473 -1.0000 -1.8793 0.0000 TT. TT. ..TT. ....TT. ......TT. ........TT. ..........TT. E2 2.0000 0.3473 -1.8793 -1.0000 1.5320 0.0000 ... ... TT... ..TT... ....TT... ......TT... ........TT... E3 2.0000 -1.0000 -1.0000 2.0000 -1.0000 0.0000 ... ... ..... TT..... ..TT..... ....TT..... TT....TT..... E4 2.0000 -1.8793 1.5320 -1.0000 0.3473 0.0000 ... ... ..... ....... TT....... TTTT....... ..TTTT....... Irrational character values: 1.532088886238 = 2*cos(2*π/9) 0.347296355334 = 2*cos(4*π/9) -1.879385241572 = 2*cos(8*π/9) Symmetry of Rotations and Cartesian products A1 p+d+f+g+h+i+j+k+2l+2m z, z^{2}, z^{3}, z^{4}, z^{5}, z^{6} A2 R+l+m R_{z} E1 R+p+d+f+g+h+i+j+2k+2l+3m {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+2j+2k+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+2i+2j+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})}, {x^{2}(x^{2}−3y^{2})^{2}−y^{2}(3x^{2}−y^{2})^{2}, xy(x^{2}−3y^{2})(3x^{2}−y^{2})}, {xz^{3}(x^{2}−3y^{2}), yz^{3}(3x^{2}−y^{2})} E4 g+2h+2i+2j+2k+2l+2m {(x^{2}−y^{2})^{2}−4x^{2}y^{2}, xy(x^{2}−y^{2})}, {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})}, {z((x^{2}−y^{2})^{2}−4x^{2}y^{2}), xyz(x^{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})}, {z^{2}((x^{2}−y^{2})^{2}−4x^{2}y^{2}), xyz^{2}(x^{2}−y^{2})} Notes: α The order of the C_{9v} point group is 18, and the order of the principal axis (C_{9}) is 9. The group has 6 irreducible representations. β The C_{9v} point group is isomorphic to D_{9}. γ The C_{9v} point group is generated by two symmetry elements, C_{9} and any σ_{v}. Also, the group may be generated from two σ_{v} planes (some pairs will yield smaller groups, though; choosing a minimum angle is safe). δ The group contains one set of symmetry planes σ_{v} intersecting in the principal (z) axis. The xz plane (but not the yz plane) is a member of that set. ε The lowest nonvanishing multipole moment in C_{9v} 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 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. θ The point group corresponds to a polygon inconstructible by the classical means of ruler and compass. Yet it becomes constructible if angle trisection is allowed, e.g., with neusis construction or origami. This is because the order of the principal axis is given by a product of any number of different Pierpont primes (...,5,7,13,17,19,37,73,97,109,163,...) times arbitrary powers of two and three. ι The regular nonagon or enneagon is not constructible by ruler and compass because cos(2*π/9) has an algebraic degree of 3. (It can be constructed by extended methods that allow angle trisection, as this corresponds to solving cubic equations). The value of cos(2*π/9) can be expressed using cubic roots and complex numbers, which, however, is not very useful for a real-valued quantity: 2*cos(2π/9) = (^{3}√−4+i*4*√3 + ^{3}√−4−i*4*√3)/2.
C_{7v} | ||
C_{8v} | ||
C_{9} | C_{9v} | C_{9h} D_{9} D_{9h} D_{9d} |
C_{10v} | ||
C_{11v} |
This Character Table for the C_{9v} point group was created by Gernot Katzer.
For other groups and some explanations, see the Main Page.