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E51 2.00000 -1.99195 1.96786 -1.92793 1.87247 -1.80194 1.71690 -1.61803 1.50614 -1.38213 1.24698 -1.10179 0.94774 -0.78605 0.61803 -0.44504 0.26847 -0.08973 -0.08973 0.26847 -0.44504 0.61803 -0.78605 0.94774 -1.10179 1.24698 -1.38213 1.50614 -1.61803 1.71690 -1.80194 1.87247 -1.92793 1.96786 -1.99195 2.00000 -1.99195 1.96786 -1.92793 1.87247 -1.80194 1.71690 -1.61803 1.50614 -1.38213 1.24698 -1.10179 0.94774 -0.78605 0.61803 -0.44504 0.26847 -0.08973 0.00000 ... ... ..... ....... ......... ........... ............. E52 2.00000 -1.99910 1.99642 -1.99195 1.98569 -1.97766 1.96786 -1.95630 1.94298 -1.92793 1.91115 -1.89265 1.87247 -1.85061 1.82709 -1.80194 1.77517 -1.74682 1.71690 -1.68544 1.65248 -1.61803 1.58214 -1.54483 1.50614 -1.46610 1.42475 -1.38213 1.33826 -1.29320 1.24698 -1.19964 1.15123 -1.10179 1.05137 -1.00000 0.94774 -0.89463 0.84071 -0.78605 0.73068 -0.67466 0.61803 -0.56085 0.50317 -0.44504 0.38651 -0.32764 0.26847 -0.20906 0.14946 -0.08973 0.02992 0.00000 ... ... ..... ....... ......... ........... ............. Irrational character values: 1.996420259535 = 2*cos(2*π/105) 1.985693852684 = 2*cos(4*π/105) 1.967859177197 = 2*cos(6*π/105) = 2*cos(2*π/35) 1.942980076586 = 2*cos(8*π/105) 1.911145611572 = 2*cos(10*π/105) = 2*cos(2*π/21) 1.872469741279 = 2*cos(12*π/105) = 2*cos(4*π/35) 1.827090915285 = 2*cos(14*π/105) = 2*cos(2*π/15) = (1+√5+√30−6*√5)/4 1.775171578009 = 2*cos(16*π/105) 1.716897587204 = 2*cos(18*π/105) = 2*cos(6*π/35) 1.652477548632 = 2*cos(20*π/105) = 2*cos(4*π/21) 1.582142069313 = 2*cos(22*π/105) 1.506142932007 = 2*cos(24*π/105) = 2*cos(8*π/35) 1.424752193903 = 2*cos(26*π/105) 1.338261212718 = 2*cos(28*π/105) = 2*cos(4*π/15) = (1−√5+√30+6*√5)/4 1.246979603717 = 2*cos(30*π/105) = 2*cos(2*π/7) 1.151234131371 = 2*cos(32*π/105) 1.051367539621 = 2*cos(34*π/105) 0.947737324946 = 2*cos(36*π/105) = 2*cos(12*π/35) 0.840714456619 = 2*cos(38*π/105) 0.730682048733 = 2*cos(40*π/105) = 2*cos(8*π/21) 0.618033988750 = 2*cos(42*π/105) = 2*cos(2*π/5) = (√5−1)/2 0.503173527489 = 2*cos(44*π/105) 0.386511835591 = 2*cos(46*π/105) 0.268466531635 = 2*cos(48*π/105) = 2*cos(16*π/35) 0.149460187173 = 2*cos(50*π/105) = 2*cos(10*π/21) 0.029918814031 = 2*cos(52*π/105) -0.089729660701 = 2*cos(54*π/105) = 2*cos(18*π/35) -0.209056926535 = 2*cos(56*π/105) = 2*cos(8*π/15) = −(√30−6*√5−√5−1)/4 -0.327635822830 = 2*cos(58*π/105) -0.445041867913 = 2*cos(60*π/105) = 2*cos(4*π/7) -0.560854778612 = 2*cos(62*π/105) -0.674659974766 = 2*cos(64*π/105) -0.786050063308 = 2*cos(66*π/105) = 2*cos(22*π/35) -0.894626296631 = 2*cos(68*π/105) -1.101793962904 = 2*cos(72*π/105) = 2*cos(24*π/35) -1.199643789376 = 2*cos(74*π/105) -1.293199202432 = 2*cos(76*π/105) -1.382125297974 = 2*cos(78*π/105) = 2*cos(26*π/35) -1.466103743660 = 2*cos(80*π/105) = 2*cos(16*π/21) -1.544833918449 = 2*cos(82*π/105) -1.618033988750 = 2*cos(84*π/105) = 2*cos(4*π/5) = −(√5+1)/2 -1.685441917308 = 2*cos(86*π/105) -1.746816401234 = 2*cos(88*π/105) -1.801937735805 = 2*cos(90*π/105) = 2*cos(6*π/7) -1.850608600948 = 2*cos(92*π/105) -1.892654767598 = 2*cos(94*π/105) -1.927925721392 = 2*cos(96*π/105) = 2*cos(32*π/35) -1.956295201468 = 2*cos(98*π/105) = 2*cos(14*π/15) = −(√5−1+√30+6*√5)/4 -1.977661652450 = 2*cos(100*π/105) = 2*cos(20*π/21) -1.991948587990 = 2*cos(102*π/105) = 2*cos(34*π/35) -1.999104864567 = 2*cos(104*π/105) Symmetry of Rotations and Cartesian products A1 d+g+i+k+m z2, z4, z6 A2 R+p+f+h+j+l Rz, z, z3, z5 E1 R+p+d+f+g+h+i+j+k+l+m {Rx, Ry}, {x, y}, {xz, yz}, {xz2, yz2}, {xz3, yz3}, {xz4, yz4}, {xz5, yz5} E2 d+f+g+h+i+j+k+l+m {x2−y2, xy}, {z(x2−y2), xyz}, {z2(x2−y2), xyz2}, {z3(x2−y2), xyz3}, {z4(x2−y2), xyz4} E3 f+g+h+i+j+k+l+m {x(x2−3y2), y(3x2−y2)}, {xz(x2−3y2), yz(3x2−y2)}, {xz2(x2−3y2), yz2(3x2−y2)}, {xz3(x2−3y2), yz3(3x2−y2)} E4 g+h+i+j+k+l+m {(x2−y2)2−4x2y2, xy(x2−y2)}, {z((x2−y2)2−4x2y2), xyz(x2−y2)}, {z2((x2−y2)2−4x2y2), xyz2(x2−y2)} E5 h+i+j+k+l+m {x(x2−(5+2√5)y2)(x2−(5−2√5)y2), y((5+2√5)x2−y2)((5−2√5)x2−y2)}, {xz(x2−(5+2√5)y2)(x2−(5−2√5)y2), yz((5+2√5)x2−y2)((5−2√5)x2−y2)} E6 i+j+k+l+m {x2(x2−3y2)2−y2(3x2−y2)2, xy(x2−3y2)(3x2−y2)} E7 j+k+l+m E8 k+l+m E9 l+m E10 m Notes: α The order of the D105 point group is 210, and the order of the principal axis (C105) is 105. The group has 54 irreducible representations. β The D105 point group is isomorphic to C105v. γ The D105 point group is generated by two symmetry elements, C105 and a perpendicular C2′. Also, the group may be generated from two C2′ axes (some pairs will yield smaller groups, though; choosing a minimum angle is safe). δ The lowest nonvanishing multipole moment in D105 is 4 (quadrupole moment). ε This point group is non-Abelian (some symmetry operations are not commutative). Therefore, the character table contains multi-membered classes and degenerate irreducible representations. ζ The point group is chiral, as it does not contain any mirroring operation. 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. While some characters of this group can be expressed using real integers and square roots alone, others need complex numbers and third roots.
D103 | ||
D104 | ||
C105 C105v C105h | D105 | D105h D105d |
D106 | ||
D107 |
This Character Table for the D105 point group was created by Gernot Katzer.
For other groups and some explanations, see the Main Page.