C15E 2 C15 2 C15^2 2 C5 2 C15^4 2 C3 2 C5^2 2 C15^7 <R> <p> <—d—> <——f——> <———g———> <————h————> <—————i—————> A 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 ..T ..T ....T ......T ........T ..........T ............T E1 * 2.0000 1.8270 1.3382 0.6180 -0.2090 -1.0000 -1.6180 -1.9563 TT. TT. ..TT. ....TT. ......TT. ........TT. ..........TT. E2 * 2.0000 1.3382 -0.2090 -1.6180 -1.9563 -1.0000 0.6180 1.8270 ... ... TT... ..TT... ....TT... ......TT... ........TT... E3 * 2.0000 0.6180 -1.6180 -1.6180 0.6180 2.0000 0.6180 -1.6180 ... ... ..... TT..... ..TT..... ....TT..... ......TT..... E4 * 2.0000 -0.2090 -1.9563 0.6180 1.8270 -1.0000 -1.6180 1.3382 ... ... ..... ....... TT....... ..TT....... ....TT....... E5 * 2.0000 -1.0000 -1.0000 2.0000 -1.0000 -1.0000 2.0000 -1.0000 ... ... ..... ....... ......... TT......... ..TT......... E6 * 2.0000 -1.6180 0.6180 0.6180 -1.6180 2.0000 -1.6180 0.6180 ... ... ..... ....... ......... ........... TT........... E7 * 2.0000 -1.9563 1.8270 -1.6180 1.3382 -1.0000 0.6180 -0.2090 ... ... ..... ....... ......... ........... ............. Irrational character values: 1.827090915285 = 2*cos(2*π/15) = (1+√5+√30−6*√5)/4 1.338261212718 = 2*cos(4*π/15) = (1−√5+√30+6*√5)/4 0.618033988750 = 2*cos(6*π/15) = 2*cos(2*π/5) = (√5−1)/2 -0.209056926535 = 2*cos(8*π/15) = −(√30−6*√5−√5−1)/4 -1.618033988750 = 2*cos(12*π/15) = 2*cos(4*π/5) = −(√5+1)/2 -1.956295201468 = 2*cos(14*π/15) = −(√5−1+√30+6*√5)/4 Symmetry of Rotations and Cartesian products A R+p+d+f+g+h+i+j+k+l+m R_{z},z,z^{2},z^{3},z^{4},z^{5},z^{6}E1 R+p+d+f+g+h+i+j+k+l+m {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+l+m {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+k+l+m {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+j+k+l+m {(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+i+j+k+l+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})}, {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})} E6 i+j+k+2l+2m {x^{2}(x^{2}−3y^{2})^{2}−y^{2}(3x^{2}−y^{2})^{2},xy(x^{2}−3y^{2})(3x^{2}−y^{2})} E7 j+2k+2l+2m Notes: α The order of the C_{15}point group is 15, and the order of the principal axis (C_{15}) is 15. The group has 8 irreducible representations. β The C_{15}point group is generated by one single symmetry element, C_{15}. Therefore, it is a cyclic group. γ The lowest nonvanishing multipole moment in C_{15}is 2 (dipole moment). δ This is an Abelian point group (the commutative law holds between all symmetry operations). The C_{15}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. ε Because the group is Abelian and the maximum order of rotation is >2, some irreducible representations have complex characters. These 14 cases have been combined into 7 two-dimensional representations that are no longer irreducible but have real-valued characters. Accordingly, 7 pairs of left and right rotations have been combined into one two-membered pseudo-class each. ζ The 7 reducible “E” representations almost behave like true irreducible representations. Their norm, however, istwicethe 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. 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 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. κ The fact that the regular pentagon is constructible is known since antiquity; Eukleides already discovered a construction for it. The double cosine of 2π/5 is equal to the reciprocal of the Golden Ratio of (1+√5)/2 = 1.61803. The regular 15-gon can be derived from the regular pentagon by overlaying it with a equilateral triangle.

C_{13} | ||

C_{14} | ||

C_{15} | C_{15v} C_{15h} D_{15} D_{15h} D_{15d}
| |

C_{16} | ||

C_{17} |

This Character Table for the **C _{15}** point group was created by Gernot Katzer.

For other groups and some explanations, see the Main Page.