C19h E 2 C19 2 C19^2 2 C19^3 2 C19^4 2 C19^5 2 C19^6 2 C19^7 2 C19^8 2 C19^9 sh 2 S19 2 S19^3 2 S19^5 2 S19^7 2 S19^9 2 S19^11 2 S19^13 2 S19^15 2 S19^17 <R> <p> <—d—> <——f——> <———g———> <————h————> <—————i—————> A' 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 ..T ... ....T ....... ........T ........... ............T A" 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 ... ..T ..... ......T ......... ..........T ............. E1' * 2.00000 1.89163 1.57828 1.09390 0.49097 -0.16516 -0.80339 -1.35456 -1.75895 -1.97272 2.00000 1.89163 1.09390 -0.16516 -1.35456 -1.97272 -1.75895 -0.80339 0.49097 1.57828 ... TT. ..... ....TT. ......... ........TT. ............. E1" * 2.00000 1.89163 1.57828 1.09390 0.49097 -0.16516 -0.80339 -1.35456 -1.75895 -1.97272 -2.00000 -1.89163 -1.09390 0.16516 1.35456 1.97272 1.75895 0.80339 -0.49097 -1.57828 TT. ... ..TT. ....... ......TT. ........... ..........TT. E2' * 2.00000 1.57828 0.49097 -0.80339 -1.75895 -1.97272 -1.35456 -0.16516 1.09390 1.89163 2.00000 1.57828 -0.80339 -1.97272 -0.16516 1.89163 1.09390 -1.35456 -1.75895 0.49097 ... ... TT... ....... ....TT... ........... ........TT... E2" * 2.00000 1.57828 0.49097 -0.80339 -1.75895 -1.97272 -1.35456 -0.16516 1.09390 1.89163 -2.00000 -1.57828 0.80339 1.97272 0.16516 -1.89163 -1.09390 1.35456 1.75895 -0.49097 ... ... ..... ..TT... ......... ......TT... ............. E3' * 2.00000 1.09390 -0.80339 -1.97272 -1.35456 0.49097 1.89163 1.57828 -0.16516 -1.75895 2.00000 1.09390 -1.97272 0.49097 1.57828 -1.75895 -0.16516 1.89163 -1.35456 -0.80339 ... ... ..... TT..... ......... ....TT..... ............. E3" * 2.00000 1.09390 -0.80339 -1.97272 -1.35456 0.49097 1.89163 1.57828 -0.16516 -1.75895 -2.00000 -1.09390 1.97272 -0.49097 -1.57828 1.75895 0.16516 -1.89163 1.35456 0.80339 ... ... ..... ....... ..TT..... ........... ......TT..... E4' * 2.00000 0.49097 -1.75895 -1.35456 1.09390 1.89163 -0.16516 -1.97272 -0.80339 1.57828 2.00000 0.49097 -1.35456 1.89163 -1.97272 1.57828 -0.80339 -0.16516 1.09390 -1.75895 ... ... ..... ....... TT....... ........... ....TT....... E4" * 2.00000 0.49097 -1.75895 -1.35456 1.09390 1.89163 -0.16516 -1.97272 -0.80339 1.57828 -2.00000 -0.49097 1.35456 -1.89163 1.97272 -1.57828 0.80339 0.16516 -1.09390 1.75895 ... ... ..... ....... ......... ..TT....... ............. E5' * 2.00000 -0.16516 -1.97272 0.49097 1.89163 -0.80339 -1.75895 1.09390 1.57828 -1.35456 2.00000 -0.16516 0.49097 -0.80339 1.09390 -1.35456 1.57828 -1.75895 1.89163 -1.97272 ... ... ..... ....... ......... TT......... ............. E5" * 2.00000 -0.16516 -1.97272 0.49097 1.89163 -0.80339 -1.75895 1.09390 1.57828 -1.35456 -2.00000 0.16516 -0.49097 0.80339 -1.09390 1.35456 -1.57828 1.75895 -1.89163 1.97272 ... ... ..... ....... ......... ........... ..TT......... E6' * 2.00000 -0.80339 -1.35456 1.89163 -0.16516 -1.75895 1.57828 0.49097 -1.97272 1.09390 2.00000 -0.80339 1.89163 -1.75895 0.49097 1.09390 -1.97272 1.57828 -0.16516 -1.35456 ... ... ..... ....... ......... ........... TT........... E6" * 2.00000 -0.80339 -1.35456 1.89163 -0.16516 -1.75895 1.57828 0.49097 -1.97272 1.09390 -2.00000 0.80339 -1.89163 1.75895 -0.49097 -1.09390 1.97272 -1.57828 0.16516 1.35456 ... ... ..... ....... ......... ........... ............. E7' * 2.00000 -1.35456 -0.16516 1.57828 -1.97272 1.09390 0.49097 -1.75895 1.89163 -0.80339 2.00000 -1.35456 1.57828 1.09390 -1.75895 -0.80339 1.89163 0.49097 -1.97272 -0.16516 ... ... ..... ....... ......... ........... ............. E7" * 2.00000 -1.35456 -0.16516 1.57828 -1.97272 1.09390 0.49097 -1.75895 1.89163 -0.80339 -2.00000 1.35456 -1.57828 -1.09390 1.75895 0.80339 -1.89163 -0.49097 1.97272 0.16516 ... ... ..... ....... ......... ........... ............. E8' * 2.00000 -1.75895 1.09390 -0.16516 -0.80339 1.57828 -1.97272 1.89163 -1.35456 0.49097 2.00000 -1.75895 -0.16516 1.57828 1.89163 0.49097 -1.35456 -1.97272 -0.80339 1.09390 ... ... ..... ....... ......... ........... ............. E8" * 2.00000 -1.75895 1.09390 -0.16516 -0.80339 1.57828 -1.97272 1.89163 -1.35456 0.49097 -2.00000 1.75895 0.16516 -1.57828 -1.89163 -0.49097 1.35456 1.97272 0.80339 -1.09390 ... ... ..... ....... ......... ........... ............. E9' * 2.00000 -1.97272 1.89163 -1.75895 1.57828 -1.35456 1.09390 -0.80339 0.49097 -0.16516 2.00000 -1.97272 -1.75895 -1.35456 -0.80339 -0.16516 0.49097 1.09390 1.57828 1.89163 ... ... ..... ....... ......... ........... ............. E9" * 2.00000 -1.97272 1.89163 -1.75895 1.57828 -1.35456 1.09390 -0.80339 0.49097 -0.16516 -2.00000 1.97272 1.75895 1.35456 0.80339 0.16516 -0.49097 -1.09390 -1.57828 -1.89163 ... ... ..... ....... ......... ........... ............. Irrational character values: 1.972722606805 = 2*cos(2*π/38) = 2*cos(π/19) 1.891634483401 = 2*cos(4*π/38) = 2*cos(2*π/19) 1.758947502413 = 2*cos(6*π/38) = 2*cos(3*π/19) 1.578281018793 = 2*cos(8*π/38) = 2*cos(4*π/19) 1.354563143251 = 2*cos(10*π/38) = 2*cos(5*π/19) 1.093896316245 = 2*cos(12*π/38) = 2*cos(6*π/19) 0.803390849306 = 2*cos(14*π/38) = 2*cos(7*π/19) 0.490970974282 = 2*cos(16*π/38) = 2*cos(8*π/19) 0.165158690945 = 2*cos(18*π/38) = 2*cos(9*π/19) Symmetry of Rotations and Cartesian products A' R+d+g+i+k+m Rz, z2, z4, z6 A" p+f+h+j+l z, z3, z5 E1' p+f+h+j+l {x, y}, {xz2, yz2}, {xz4, yz4} E1" R+d+g+i+k+m {Rx, Ry}, {xz, yz}, {xz3, yz3}, {xz5, yz5} E2' d+g+i+k+m {x2−y2, xy}, {z2(x2−y2), xyz2}, {z4(x2−y2), xyz4} E2" f+h+j+l {z(x2−y2), xyz}, {z3(x2−y2), xyz3} E3' f+h+j+l {x(x2−3y2), y(3x2−y2)}, {xz2(x2−3y2), yz2(3x2−y2)} E3" g+i+k+m {xz(x2−3y2), yz(3x2−y2)}, {xz3(x2−3y2), yz3(3x2−y2)} E4' g+i+k+m {(x2−y2)2−4x2y2, xy(x2−y2)}, {z2((x2−y2)2−4x2y2), xyz2(x2−y2)} E4" h+j+l {z((x2−y2)2−4x2y2), xyz(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)} E5" 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)} E6' i+k+m {x2(x2−3y2)2−y2(3x2−y2)2, xy(x2−3y2)(3x2−y2)} E6" j+l E7' j+l E7" k+m E8' k+m E8" l E9' l+m E9" m Notes: α The order of the C19h point group is 38, and the order of the principal axis (S19) is 38. The group has 20 irreducible representations. β The C19h point group could also be named S19, as it contains the S19 axis as its only symmetry element. Another rare designation is C38i because the S19 axis is identical to a roto-inversion axis of order 38. γ The C19h point group is isomorphic to C38 and S38. δ The C19h point group is generated by one single symmetry element, S19. Therefore, it is a cyclic group. The canonical choice, however, is to use redundant generators: C19 and σh. ε The lowest nonvanishing multipole moment in C19h is 4 (quadrupole moment). ζ This is an Abelian point group (the commutative law holds between all symmetry operations). The C19h 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 36 cases have been combined into 18 two-dimensional representations that are no longer irreducible but have real-valued characters. Accordingly, 18 pairs of left and right rotations have been combined into one two-membered pseudo-class each. θ The 18 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 just 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. All characters of this group can be expressed using complex numbers, elementary arithmetic operations, square roots and third roots.
| C17h | ||
| C18h | ||
| C19 C19v | C19h | D19 D19h D19d |
| C20h | ||
| C21h |
This Character Table for the C19h point group was created by Gernot Katzer.
For other groups and some explanations, see the Main Page.