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E51 2.00000 -1.78201 1.17557 -0.31287 -0.61803 1.41421 -1.90211 1.97538 -1.61803 0.90798 0.00000 -0.90798 1.61803 -1.97538 1.90211 -1.41421 0.61803 0.31287 -1.17557 1.78201 -2.00000 1.78201 -1.17557 0.31287 0.61803 -1.41421 1.90211 -1.97538 1.61803 -0.90798 0.00000 0.90798 -1.61803 1.97538 -1.90211 1.41421 -0.61803 -0.31287 1.17557 -1.78201 2.00000 -1.78201 1.17557 -0.31287 -0.61803 1.41421 -1.90211 1.97538 -1.61803 0.90798 0.00000 -0.90798 1.61803 -1.97538 1.90211 -1.41421 0.61803 0.31287 -1.17557 1.78201 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E52 2.00000 -1.82709 1.33826 -0.61803 -0.20906 1.00000 -1.61803 1.95630 -1.95630 1.61803 -1.00000 0.20906 0.61803 -1.33826 1.82709 -2.00000 1.82709 -1.33826 0.61803 0.20906 -1.00000 1.61803 -1.95630 1.95630 -1.61803 1.00000 -0.20906 -0.61803 1.33826 -1.82709 2.00000 -1.82709 1.33826 -0.61803 -0.20906 1.00000 -1.61803 1.95630 -1.95630 1.61803 -1.00000 0.20906 0.61803 -1.33826 1.82709 -2.00000 1.82709 -1.33826 0.61803 0.20906 -1.00000 1.61803 -1.95630 1.95630 -1.61803 1.00000 -0.20906 -0.61803 1.33826 -1.82709 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E53 2.00000 -1.86716 1.48629 -0.90798 0.20906 0.51764 -1.17557 1.67734 -1.95630 1.97538 -1.73205 1.25864 -0.61803 -0.10467 0.81347 -1.41421 1.82709 -1.99726 1.90211 -1.55429 1.00000 -0.31287 -0.41582 1.08928 -1.61803 1.93185 -1.98904 1.78201 -1.33826 0.71674 0.00000 -0.71674 1.33826 -1.78201 1.98904 -1.93185 1.61803 -1.08928 0.41582 0.31287 -1.00000 1.55429 -1.90211 1.99726 -1.82709 1.41421 -0.81347 0.10467 0.61803 -1.25864 1.73205 -1.97538 1.95630 -1.67734 1.17557 -0.51764 -0.20906 0.90798 -1.48629 1.86716 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E54 2.00000 -1.90211 1.61803 -1.17557 0.61803 0.00000 -0.61803 1.17557 -1.61803 1.90211 -2.00000 1.90211 -1.61803 1.17557 -0.61803 0.00000 0.61803 -1.17557 1.61803 -1.90211 2.00000 -1.90211 1.61803 -1.17557 0.61803 0.00000 -0.61803 1.17557 -1.61803 1.90211 -2.00000 1.90211 -1.61803 1.17557 -0.61803 0.00000 0.61803 -1.17557 1.61803 -1.90211 2.00000 -1.90211 1.61803 -1.17557 0.61803 0.00000 -0.61803 1.17557 -1.61803 1.90211 -2.00000 1.90211 -1.61803 1.17557 -0.61803 0.00000 0.61803 -1.17557 1.61803 -1.90211 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E55 2.00000 -1.93185 1.73205 -1.41421 1.00000 -0.51764 0.00000 0.51764 -1.00000 1.41421 -1.73205 1.93185 -2.00000 1.93185 -1.73205 1.41421 -1.00000 0.51764 0.00000 -0.51764 1.00000 -1.41421 1.73205 -1.93185 2.00000 -1.93185 1.73205 -1.41421 1.00000 -0.51764 0.00000 0.51764 -1.00000 1.41421 -1.73205 1.93185 -2.00000 1.93185 -1.73205 1.41421 -1.00000 0.51764 0.00000 -0.51764 1.00000 -1.41421 1.73205 -1.93185 2.00000 -1.93185 1.73205 -1.41421 1.00000 -0.51764 0.00000 0.51764 -1.00000 1.41421 -1.73205 1.93185 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ..TT......... E56 2.00000 -1.95630 1.82709 -1.61803 1.33826 -1.00000 0.61803 -0.20906 -0.20906 0.61803 -1.00000 1.33826 -1.61803 1.82709 -1.95630 2.00000 -1.95630 1.82709 -1.61803 1.33826 -1.00000 0.61803 -0.20906 -0.20906 0.61803 -1.00000 1.33826 -1.61803 1.82709 -1.95630 2.00000 -1.95630 1.82709 -1.61803 1.33826 -1.00000 0.61803 -0.20906 -0.20906 0.61803 -1.00000 1.33826 -1.61803 1.82709 -1.95630 2.00000 -1.95630 1.82709 -1.61803 1.33826 -1.00000 0.61803 -0.20906 -0.20906 0.61803 -1.00000 1.33826 -1.61803 1.82709 -1.95630 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ..TT....... ............. E57 2.00000 -1.97538 1.90211 -1.78201 1.61803 -1.41421 1.17557 -0.90798 0.61803 -0.31287 0.00000 0.31287 -0.61803 0.90798 -1.17557 1.41421 -1.61803 1.78201 -1.90211 1.97538 -2.00000 1.97538 -1.90211 1.78201 -1.61803 1.41421 -1.17557 0.90798 -0.61803 0.31287 0.00000 -0.31287 0.61803 -0.90798 1.17557 -1.41421 1.61803 -1.78201 1.90211 -1.97538 2.00000 -1.97538 1.90211 -1.78201 1.61803 -1.41421 1.17557 -0.90798 0.61803 -0.31287 0.00000 0.31287 -0.61803 0.90798 -1.17557 1.41421 -1.61803 1.78201 -1.90211 1.97538 -2.00000 0.00000 0.00000 ... ... ..... ....... ..TT..... ........... ......TT..... E58 2.00000 -1.98904 1.95630 -1.90211 1.82709 -1.73205 1.61803 -1.48629 1.33826 -1.17557 1.00000 -0.81347 0.61803 -0.41582 0.20906 0.00000 -0.20906 0.41582 -0.61803 0.81347 -1.00000 1.17557 -1.33826 1.48629 -1.61803 1.73205 -1.82709 1.90211 -1.95630 1.98904 -2.00000 1.98904 -1.95630 1.90211 -1.82709 1.73205 -1.61803 1.48629 -1.33826 1.17557 -1.00000 0.81347 -0.61803 0.41582 -0.20906 0.00000 0.20906 -0.41582 0.61803 -0.81347 1.00000 -1.17557 1.33826 -1.48629 1.61803 -1.73205 1.82709 -1.90211 1.95630 -1.98904 2.00000 0.00000 0.00000 ... ... ..... ..TT... ......... ......TT... ............. E59 2.00000 -1.99726 1.98904 -1.97538 1.95630 -1.93185 1.90211 -1.86716 1.82709 -1.78201 1.73205 -1.67734 1.61803 -1.55429 1.48629 -1.41421 1.33826 -1.25864 1.17557 -1.08928 1.00000 -0.90798 0.81347 -0.71674 0.61803 -0.51764 0.41582 -0.31287 0.20906 -0.10467 0.00000 0.10467 -0.20906 0.31287 -0.41582 0.51764 -0.61803 0.71674 -0.81347 0.90798 -1.00000 1.08928 -1.17557 1.25864 -1.33826 1.41421 -1.48629 1.55429 -1.61803 1.67734 -1.73205 1.78201 -1.82709 1.86716 -1.90211 1.93185 -1.95630 1.97538 -1.98904 1.99726 -2.00000 0.00000 0.00000 TT. ... ..TT. ....... ......TT. ........... ..........TT. Irrational character values: 1.997259069509 = 2*cos(2*π/120) = 2*cos(π/60) = (√2−√6−√10+√30+2*(1+√3)*(√5+√5))/8 1.989043790737 = 2*cos(4*π/120) = 2*cos(π/30) = (√15+√3+√10−2*√5)/4 1.975376681190 = 2*cos(6*π/120) = 2*cos(π/20) = (√2+√10+2*(√5−√5))/4 1.956295201468 = 2*cos(8*π/120) = 2*cos(π/15) = (√5−1+√30+6*√5)/4 1.931851652578 = 2*cos(10*π/120) = 2*cos(π/12) = √2+√3 1.902113032590 = 2*cos(12*π/120) = 2*cos(π/10) = (√10+2*√5)/2 1.867160852994 = 2*cos(14*π/120) = 2*cos(7*π/60) = (√2+√6+√10+√30−2*(1−√3)*(√5−√5))/8 1.827090915285 = 2*cos(16*π/120) = 2*cos(2*π/15) = (1+√5+√30−6*√5)/4 1.782013048377 = 2*cos(18*π/120) = 2*cos(3*π/20) = (√10−√2+2*√5+√5)/4 1.732050807569 = 2*cos(20*π/120) = 2*cos(π/6) = √3 1.677341135891 = 2*cos(22*π/120) = 2*cos(11*π/60) = (√6−√2+√10−√30+2*(1+√3)*(√5+√5))/8 1.618033988750 = 2*cos(24*π/120) = 2*cos(π/5) = (√5+1)/2 1.554291922914 = 2*cos(26*π/120) = 2*cos(13*π/60) = (√6−√2−√10+√30+2*(1+√3)*(√5−√5))/8 1.486289650955 = 2*cos(28*π/120) = 2*cos(7*π/30) = (√15−√3+√10+2*√5)/4 1.414213562373 = 2*cos(30*π/120) = 2*cos(π/4) = √2 1.338261212718 = 2*cos(32*π/120) = 2*cos(4*π/15) = (1−√5+√30+6*√5)/4 1.258640782100 = 2*cos(34*π/120) = 2*cos(17*π/60) = (√2+√6+√10+√30+2*(1−√3)*(√5−√5))/8 1.175570504585 = 2*cos(36*π/120) = 2*cos(3*π/10) = (√10−2*√5)/2 1.089278070030 = 2*cos(38*π/120) = 2*cos(19*π/60) = (√10−√2−√6+√30−2*(1−√3)*(√5+√5))/8 0.907980999479 = 2*cos(42*π/120) = 2*cos(7*π/20) = (√2−√10+2*√5+√5)/4 0.813473286152 = 2*cos(44*π/120) = 2*cos(11*π/30) = (√15+√3−√10−2*√5)/4 0.716735899091 = 2*cos(46*π/120) = 2*cos(23*π/60) = (√2−√6+√10−√30+2*(1+√3)*(√5−√5))/8 0.618033988750 = 2*cos(48*π/120) = 2*cos(2*π/5) = (√5−1)/2 0.517638090205 = 2*cos(50*π/120) = 2*cos(5*π/12) = √2−√3 0.415823381636 = 2*cos(52*π/120) = 2*cos(13*π/30) = (√3−√15+√10+2*√5)/4 0.312868930080 = 2*cos(54*π/120) = 2*cos(9*π/20) = (√2+√10−2*(√5−√5))/4 0.209056926535 = 2*cos(56*π/120) = 2*cos(7*π/15) = (√30−6*√5−√5−1)/4 0.104671912486 = 2*cos(58*π/120) = 2*cos(29*π/60) = (√10−√2−√6+√30+2*(1−√3)*(√5+√5))/8 Symmetry of Rotations and Cartesian products A1 d+g+i+k+m z2, z4, z6 A2 R Rz B2 p+f+h+j+l z, z3, z5 E1 p+f+h+j+l {x, y}, {xz2, yz2}, {xz4, yz4} E2 d+g+i+k+m {x2−y2, xy}, {z2(x2−y2), xyz2}, {z4(x2−y2), xyz4} E3 f+h+j+l {x(x2−3y2), y(3x2−y2)}, {xz2(x2−3y2), yz2(3x2−y2)} E4 g+i+k+m {(x2−y2)2−4x2y2, xy(x2−y2)}, {z2((x2−y2)2−4x2y2), xyz2(x2−y2)} E5 h+j+l {x(x2−(5+2√5)y2)(x2−(5−2√5)y2), y((5+2√5)x2−y2)((5−2√5)x2−y2)} E6 i+k+m {x2(x2−3y2)2−y2(3x2−y2)2, xy(x2−3y2)(3x2−y2)} E7 j+l E8 k+m E9 l E10 m E51 m E52 l E53 k+m E54 j+l E55 i+k+m {xz(x2−(5+2√5)y2)(x2−(5−2√5)y2), yz((5+2√5)x2−y2)((5−2√5)x2−y2)} E56 h+j+l {z((x2−y2)2−4x2y2), xyz(x2−y2)} E57 g+i+k+m {xz(x2−3y2), yz(3x2−y2)}, {xz3(x2−3y2), yz3(3x2−y2)} E58 f+h+j+l {z(x2−y2), xyz}, {z3(x2−y2), xyz3} E59 R+d+g+i+k+m {Rx, Ry}, {xz, yz}, {xz3, yz3}, {xz5, yz5} Notes: α The order of the D60d point group is 240, and the order of the principal axis (S120) is 120. The group has 63 irreducible representations. β The D60d point group is isomorphic to C120v and D120. γ The D60d point group is generated by two symmetry elements, S120 and either a perpendicular C2′ or a vertical σd. Also, the group may be generated from a C2′ plus a σd (some pairs will yield smaller groups, though; choosing a minimum angle is safe). δ The group contains one set of twofold symmetry axes (C2′) perpendicular to the principal (z) axis. Both x and y axes are members of that set. ε The single σd set of symmetry planes contains neither the xz nor the yz planes; but it contains the median plane (x+y)z. ζ The lowest nonvanishing multipole moment in D60d 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. 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. Regular polygons of order 30,60,120 etc. are derived from the pentagon by overlaying it with a triangle and successive halving of angles.
D58d | ||
D59d | ||
C60 C60v C60h D60 D60h | D60d | S60 |
D61d | ||
D62d |
This Character Table for the D60d point group was created by Gernot Katzer.
For other groups and some explanations, see the Main Page.