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E51 2.00000 -1.89163 1.57828 -1.09390 0.49097 0.16516 -0.80339 1.35456 -1.75895 1.97272 -1.97272 1.75895 -1.35456 0.80339 -0.16516 -0.49097 1.09390 -1.57828 1.89163 -2.00000 1.89163 -1.57828 1.09390 -0.49097 -0.16516 0.80339 -1.35456 1.75895 -1.97272 1.97272 -1.75895 1.35456 -0.80339 0.16516 0.49097 -1.09390 1.57828 -1.89163 2.00000 -1.89163 1.57828 -1.09390 0.49097 0.16516 -0.80339 1.35456 -1.75895 1.97272 -1.97272 1.75895 -1.35456 0.80339 -0.16516 -0.49097 1.09390 -1.57828 1.89163 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E52 2.00000 -1.92454 1.70384 -1.35456 0.90307 -0.38342 -0.16516 0.70128 -1.18447 1.57828 -1.85299 1.98786 -1.97272 1.80871 -1.50821 1.09390 -0.59703 0.05511 0.49097 -1.00000 1.43357 -1.75895 1.95159 -1.99696 1.89163 -1.64356 1.27145 -0.80339 0.27471 0.27471 -0.80339 1.27145 -1.64356 1.89163 -1.99696 1.95159 -1.75895 1.43357 -1.00000 0.49097 0.05511 -0.59703 1.09390 -1.50821 1.80871 -1.97272 1.98786 -1.85299 1.57828 -1.18447 0.70128 -0.16516 -0.38342 0.90307 -1.35456 1.70384 -1.92454 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E53 2.00000 -1.95159 1.80871 -1.57828 1.27145 -0.90307 0.49097 -0.05511 -0.38342 0.80339 -1.18447 1.50821 -1.75895 1.92454 -1.99696 1.97272 -1.85299 1.64356 -1.35456 1.00000 -0.59703 0.16516 0.27471 -0.70128 1.09390 -1.43357 1.70384 -1.89163 1.98786 -1.98786 1.89163 -1.70384 1.43357 -1.09390 0.70128 -0.27471 -0.16516 0.59703 -1.00000 1.35456 -1.64356 1.85299 -1.97272 1.99696 -1.92454 1.75895 -1.50821 1.18447 -0.80339 0.38342 0.05511 -0.49097 0.90307 -1.27145 1.57828 -1.80871 1.95159 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E54 2.00000 -1.97272 1.89163 -1.75895 1.57828 -1.35456 1.09390 -0.80339 0.49097 -0.16516 -0.16516 0.49097 -0.80339 1.09390 -1.35456 1.57828 -1.75895 1.89163 -1.97272 2.00000 -1.97272 1.89163 -1.75895 1.57828 -1.35456 1.09390 -0.80339 0.49097 -0.16516 -0.16516 0.49097 -0.80339 1.09390 -1.35456 1.57828 -1.75895 1.89163 -1.97272 2.00000 -1.97272 1.89163 -1.75895 1.57828 -1.35456 1.09390 -0.80339 0.49097 -0.16516 -0.16516 0.49097 -0.80339 1.09390 -1.35456 1.57828 -1.75895 1.89163 -1.97272 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E55 2.00000 -1.98786 1.95159 -1.89163 1.80871 -1.70384 1.57828 -1.43357 1.27145 -1.09390 0.90307 -0.70128 0.49097 -0.27471 0.05511 0.16516 -0.38342 0.59703 -0.80339 1.00000 -1.18447 1.35456 -1.50821 1.64356 -1.75895 1.85299 -1.92454 1.97272 -1.99696 1.99696 -1.97272 1.92454 -1.85299 1.75895 -1.64356 1.50821 -1.35456 1.18447 -1.00000 0.80339 -0.59703 0.38342 -0.16516 -0.05511 0.27471 -0.49097 0.70128 -0.90307 1.09390 -1.27145 1.43357 -1.57828 1.70384 -1.80871 1.89163 -1.95159 1.98786 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E56 2.00000 -1.99696 1.98786 -1.97272 1.95159 -1.92454 1.89163 -1.85299 1.80871 -1.75895 1.70384 -1.64356 1.57828 -1.50821 1.43357 -1.35456 1.27145 -1.18447 1.09390 -1.00000 0.90307 -0.80339 0.70128 -0.59703 0.49097 -0.38342 0.27471 -0.16516 0.05511 0.05511 -0.16516 0.27471 -0.38342 0.49097 -0.59703 0.70128 -0.80339 0.90307 -1.00000 1.09390 -1.18447 1.27145 -1.35456 1.43357 -1.50821 1.57828 -1.64356 1.70384 -1.75895 1.80871 -1.85299 1.89163 -1.92454 1.95159 -1.97272 1.98786 -1.99696 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. Irrational character values: 1.996963032867 = 2*cos(2*π/114) = 2*cos(π/57) 1.987861354636 = 2*cos(4*π/114) = 2*cos(2*π/57) 1.972722606805 = 2*cos(6*π/114) = 2*cos(π/19) 1.951592765255 = 2*cos(8*π/114) = 2*cos(4*π/57) 1.924536000619 = 2*cos(10*π/114) = 2*cos(5*π/57) 1.891634483401 = 2*cos(12*π/114) = 2*cos(2*π/19) 1.852988134430 = 2*cos(14*π/114) = 2*cos(7*π/57) 1.808714321395 = 2*cos(16*π/114) = 2*cos(8*π/57) 1.758947502413 = 2*cos(18*π/114) = 2*cos(3*π/19) 1.703838817677 = 2*cos(20*π/114) = 2*cos(10*π/57) 1.643555630450 = 2*cos(22*π/114) = 2*cos(11*π/57) 1.578281018793 = 2*cos(24*π/114) = 2*cos(4*π/19) 1.508213219554 = 2*cos(26*π/114) = 2*cos(13*π/57) 1.433565026337 = 2*cos(28*π/114) = 2*cos(14*π/57) 1.354563143251 = 2*cos(30*π/114) = 2*cos(5*π/19) 1.271447496420 = 2*cos(32*π/114) = 2*cos(16*π/57) 1.184470505330 = 2*cos(34*π/114) = 2*cos(17*π/57) 1.093896316245 = 2*cos(36*π/114) = 2*cos(6*π/19) 0.903066716622 = 2*cos(40*π/114) = 2*cos(20*π/57) 0.803390849306 = 2*cos(42*π/114) = 2*cos(7*π/19) 0.701275110386 = 2*cos(44*π/114) = 2*cos(22*π/57) 0.597029622003 = 2*cos(46*π/114) = 2*cos(23*π/57) 0.490970974282 = 2*cos(48*π/114) = 2*cos(8*π/19) 0.383421263847 = 2*cos(50*π/114) = 2*cos(25*π/57) 0.274707115637 = 2*cos(52*π/114) = 2*cos(26*π/57) 0.165158690945 = 2*cos(54*π/114) = 2*cos(9*π/19) 0.055108684736 = 2*cos(56*π/114) = 2*cos(28*π/57) 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 D114 point group is 228, and the order of the principal axis (C114) is 114. The group has 60 irreducible representations. β The D114 point group is isomorphic to D57d, D57h and C114v. γ The D114 point group is generated by two symmetry elements, C114 and a perpendicular C2′ (or, non-canonically, C2″). Also, the group may be generated from a C2′ plus a C2″ (some pairs will yield smaller groups, though; choosing a minimum angle is safe). δ There are two different sets of twofold symmetry axes perpendicular to the principal axis (z axis in standard orientation). By convention, the set denoted as C2′ has the x axis as a member, while the y axis is a member of the C2″ set. ε The lowest nonvanishing multipole moment in D114 is 4 (quadrupole 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 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 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. All characters of this group can be expressed using complex numbers, elementary arithmetic operations, square roots and third roots.
| D112 | ||
| D113 | ||
| C114 C114v C114h | D114 | D114h D114d S114 |
| D115 | ||
| D116 |
This Character Table for the D114 point group was created by Gernot Katzer.
For other groups and some explanations, see the Main Page.