C9h E 2 C9 2 C9^2 2 C3 2 C9^4 sh 2 S9 2 S3 2 S9^5 2 S9^7 <R> <p> <—d—> <——f——> <———g———> <————h————> <—————i—————> A' 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 ..T ... ....T ....... ........T ........... ............T A" 1.0000 1.0000 1.0000 1.0000 1.0000 -1.0000 -1.0000 -1.0000 -1.0000 -1.0000 ... ..T ..... ......T ......... ..........T ............. E1' * 2.0000 1.5320 0.3473 -1.0000 -1.8793 2.0000 1.5320 -1.0000 -1.8793 0.3473 ... TT. ..... ....TT. ......... ........TT. ............. E1" * 2.0000 1.5320 0.3473 -1.0000 -1.8793 -2.0000 -1.5320 1.0000 1.8793 -0.3473 TT. ... ..TT. ....... ......TT. ........... ..........TT. E2' * 2.0000 0.3473 -1.8793 -1.0000 1.5320 2.0000 0.3473 -1.0000 1.5320 -1.8793 ... ... TT... ....... ....TT... ........... ........TT... E2" * 2.0000 0.3473 -1.8793 -1.0000 1.5320 -2.0000 -0.3473 1.0000 -1.5320 1.8793 ... ... ..... ..TT... ......... ......TT... ............. E3' * 2.0000 -1.0000 -1.0000 2.0000 -1.0000 2.0000 -1.0000 2.0000 -1.0000 -1.0000 ... ... ..... TT..... ......... ....TT..... TT........... E3" * 2.0000 -1.0000 -1.0000 2.0000 -1.0000 -2.0000 1.0000 -2.0000 1.0000 1.0000 ... ... ..... ....... ..TT..... ........... ......TT..... E4' * 2.0000 -1.8793 1.5320 -1.0000 0.3473 2.0000 -1.8793 -1.0000 0.3473 1.5320 ... ... ..... ....... TT....... TT......... ....TT....... E4" * 2.0000 -1.8793 1.5320 -1.0000 0.3473 -2.0000 1.8793 1.0000 -0.3473 -1.5320 ... ... ..... ....... ......... ..TT....... ..TT......... Irrational character values: 1.879385241572 = 2*cos(2*π/18) = 2*cos(π/9) 1.532088886238 = 2*cos(4*π/18) = 2*cos(2*π/9) 0.347296355334 = 2*cos(8*π/18) = 2*cos(4*π/9) Symmetry of Rotations and Cartesian products A' R+d+g+i+k+2l+m R_{z}, z^{2}, z^{4}, z^{6} A" p+f+h+j+l+2m z, z^{3}, z^{5} E1' p+f+h+j+k+l+2m {x, y}, {xz^{2}, yz^{2}}, {xz^{4}, yz^{4}} E1" R+d+g+i+k+l+m {R_{x}, R_{y}}, {xz, yz}, {xz^{3}, yz^{3}}, {xz^{5}, yz^{5}} E2' d+g+i+j+k+l+m {x^{2}−y^{2}, xy}, {z^{2}(x^{2}−y^{2}), xyz^{2}}, {z^{4}(x^{2}−y^{2}), xyz^{4}} E2" f+h+j+k+l+m {z(x^{2}−y^{2}), xyz}, {z^{3}(x^{2}−y^{2}), xyz^{3}} E3' f+h+i+j+k+l+m {x(x^{2}−3y^{2}), y(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})} E3" g+i+j+k+l+m {xz(x^{2}−3y^{2}), yz(3x^{2}−y^{2})}, {xz^{3}(x^{2}−3y^{2}), yz^{3}(3x^{2}−y^{2})} E4' g+h+i+j+k+l+m {(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^{2}((x^{2}−y^{2})^{2}−4x^{2}y^{2}), xyz^{2}(x^{2}−y^{2})} E4" h+i+j+k+l+m {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})} Notes: α The order of the C_{9h} point group is 18, and the order of the principal axis (S_{9}) is 18. The group has 10 irreducible representations. β The C_{9h} point group could also be named S_{9}, as it contains the S_{9} axis as its only symmetry element. Another rare designation is C_{18i} because the S_{9} axis is identical to a roto-inversion axis of order 18. γ The C_{9h} point group is isomorphic to C_{18} and S_{18}. δ The C_{9h} point group is generated by one single symmetry element, S_{9}. Therefore, it is a cyclic group. The canonical choice, however, is to use redundant generators: C_{9} and σ_{h}. ε The lowest nonvanishing multipole moment in C_{9h} is 4 (quadrupole moment). ζ This is an Abelian point group (the commutative law holds between all symmetry operations). The C_{9h} group is Abelian because it contains only one symmetry element, all the powers of which necessarily commute (sufficient condition). In Abelian groups, all symmetry operations form a class of their own, and all irreducible representations are one-dimensional. η Because the group is Abelian and the maximum order of rotation is >2, some irreducible representations have complex characters. These 16 cases have been combined into 8 two-dimensional representations that are no longer irreducible but have real-valued characters. Accordingly, 8 pairs of left and right rotations have been combined into one two-membered pseudo-class each. θ The 8 reducible “E” representations almost behave like true irreducible representations. Their norm, however, is twice the group order. Therefore, they have been marked with an asterisk in the table. This is essential when trying to decompose a reducible representation into “irreducible” ones using the familiar projection formula. ι 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. κ 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. λ 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_{7h} | ||
C_{8h} | ||
C_{9} C_{9v} | C_{9h} | D_{9} D_{9h} D_{9d} |
C_{10h} | ||
C_{11h} |
This Character Table for the C_{9h} point group was created by Gernot Katzer.
For other groups and some explanations, see the Main Page.