C51 E 2 C51 2 C51^2 2 C17 2 C51^4 2 C51^5 2 C17^2 2 C51^7 2 C51^8 2 C17^3 2 C51^10 2 C51^11 2 C17^4 2 C51^13 2 C51^14 2 C17^5 2 C51^16 2 C3 2 C17^6 2 C51^19 2 C51^20 2 C17^7 2 C51^22 2 C51^23 2 C17^8 2 C51^25 <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 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 ..T ..T ....T ......T ........T ..........T ............T E1 * 2.00000 1.98484 1.93959 1.86494 1.76202 1.63239 1.47802 1.30124 1.10473 0.89148 0.66471 0.42787 0.18454 -0.06159 -0.30678 -0.54733 -0.77957 -1.00000 -1.20527 -1.39227 -1.55816 -1.70043 -1.81693 -1.90588 -1.96595 -1.99621 TT. TT. ..TT. ....TT. ......TT. ........TT. ..........TT. E2 * 2.00000 1.93959 1.76202 1.47802 1.10473 0.66471 0.18454 -0.30678 -0.77957 -1.20527 -1.55816 -1.81693 -1.96595 -1.99621 -1.90588 -1.70043 -1.39227 -1.00000 -0.54733 -0.06159 0.42787 0.89148 1.30124 1.63239 1.86494 1.98484 ... ... TT... ..TT... ....TT... ......TT... ........TT... E3 * 2.00000 1.86494 1.47802 0.89148 0.18454 -0.54733 -1.20527 -1.70043 -1.96595 -1.96595 -1.70043 -1.20527 -0.54733 0.18454 0.89148 1.47802 1.86494 2.00000 1.86494 1.47802 0.89148 0.18454 -0.54733 -1.20527 -1.70043 -1.96595 ... ... ..... TT..... ..TT..... ....TT..... ......TT..... E4 * 2.00000 1.76202 1.10473 0.18454 -0.77957 -1.55816 -1.96595 -1.90588 -1.39227 -0.54733 0.42787 1.30124 1.86494 1.98484 1.63239 0.89148 -0.06159 -1.00000 -1.70043 -1.99621 -1.81693 -1.20527 -0.30678 0.66471 1.47802 1.93959 ... ... ..... ....... TT....... ..TT....... ....TT....... E5 * 2.00000 1.63239 0.66471 -0.54733 -1.55816 -1.99621 -1.70043 -0.77957 0.42787 1.47802 1.98484 1.76202 0.89148 -0.30678 -1.39227 -1.96595 -1.81693 -1.00000 0.18454 1.30124 1.93959 1.86494 1.10473 -0.06159 -1.20527 -1.90588 ... ... ..... ....... ......... TT......... ..TT......... E6 * 2.00000 1.47802 0.18454 -1.20527 -1.96595 -1.70043 -0.54733 0.89148 1.86494 1.86494 0.89148 -0.54733 -1.70043 -1.96595 -1.20527 0.18454 1.47802 2.00000 1.47802 0.18454 -1.20527 -1.96595 -1.70043 -0.54733 0.89148 1.86494 ... ... ..... ....... ......... ........... TT........... E7 * 2.00000 1.30124 -0.30678 -1.70043 -1.90588 -0.77957 0.89148 1.93959 1.63239 0.18454 -1.39227 -1.99621 -1.20527 0.42787 1.76202 1.86494 0.66471 -1.00000 -1.96595 -1.55816 -0.06159 1.47802 1.98484 1.10473 -0.54733 -1.81693 ... ... ..... ....... ......... ........... ............. E8 * 2.00000 1.10473 -0.77957 -1.96595 -1.39227 0.42787 1.86494 1.63239 -0.06159 -1.70043 -1.81693 -0.30678 1.47802 1.93959 0.66471 -1.20527 -1.99621 -1.00000 0.89148 1.98484 1.30124 -0.54733 -1.90588 -1.55816 0.18454 1.76202 ... ... ..... ....... ......... ........... ............. E9 * 2.00000 0.89148 -1.20527 -1.96595 -0.54733 1.47802 1.86494 0.18454 -1.70043 -1.70043 0.18454 1.86494 1.47802 -0.54733 -1.96595 -1.20527 0.89148 2.00000 0.89148 -1.20527 -1.96595 -0.54733 1.47802 1.86494 0.18454 -1.70043 ... ... ..... ....... ......... ........... ............. E10 * 2.00000 0.66471 -1.55816 -1.70043 0.42787 1.98484 0.89148 -1.39227 -1.81693 0.18454 1.93959 1.10473 -1.20527 -1.90588 -0.06159 1.86494 1.30124 -1.00000 -1.96595 -0.30678 1.76202 1.47802 -0.77957 -1.99621 -0.54733 1.63239 ... ... ..... ....... ......... ........... ............. E11 * 2.00000 0.42787 -1.81693 -1.20527 1.30124 1.76202 -0.54733 -1.99621 -0.30678 1.86494 1.10473 -1.39227 -1.70043 0.66471 1.98484 0.18454 -1.90588 -1.00000 1.47802 1.63239 -0.77957 -1.96595 -0.06159 1.93959 0.89148 -1.55816 ... ... ..... ....... ......... ........... ............. E12 * 2.00000 0.18454 -1.96595 -0.54733 1.86494 0.89148 -1.70043 -1.20527 1.47802 1.47802 -1.20527 -1.70043 0.89148 1.86494 -0.54733 -1.96595 0.18454 2.00000 0.18454 -1.96595 -0.54733 1.86494 0.89148 -1.70043 -1.20527 1.47802 ... ... ..... ....... ......... ........... ............. E13 * 2.00000 -0.06159 -1.99621 0.18454 1.98484 -0.30678 -1.96595 0.42787 1.93959 -0.54733 -1.90588 0.66471 1.86494 -0.77957 -1.81693 0.89148 1.76202 -1.00000 -1.70043 1.10473 1.63239 -1.20527 -1.55816 1.30124 1.47802 -1.39227 ... ... ..... ....... ......... ........... ............. E14 * 2.00000 -0.30678 -1.90588 0.89148 1.63239 -1.39227 -1.20527 1.76202 0.66471 -1.96595 -0.06159 1.98484 -0.54733 -1.81693 1.10473 1.47802 -1.55816 -1.00000 1.86494 0.42787 -1.99621 0.18454 1.93959 -0.77957 -1.70043 1.30124 ... ... ..... ....... ......... ........... ............. E15 * 2.00000 -0.54733 -1.70043 1.47802 0.89148 -1.96595 0.18454 1.86494 -1.20527 -1.20527 1.86494 0.18454 -1.96595 0.89148 1.47802 -1.70043 -0.54733 2.00000 -0.54733 -1.70043 1.47802 0.89148 -1.96595 0.18454 1.86494 -1.20527 ... ... ..... ....... ......... ........... ............. E16 * 2.00000 -0.77957 -1.39227 1.86494 -0.06159 -1.81693 1.47802 0.66471 -1.99621 0.89148 1.30124 -1.90588 0.18454 1.76202 -1.55816 -0.54733 1.98484 -1.00000 -1.20527 1.93959 -0.30678 -1.70043 1.63239 0.42787 -1.96595 1.10473 ... ... ..... ....... ......... ........... ............. E17 * 2.00000 -1.00000 -1.00000 2.00000 -1.00000 -1.00000 2.00000 -1.00000 -1.00000 2.00000 -1.00000 -1.00000 2.00000 -1.00000 -1.00000 2.00000 -1.00000 -1.00000 2.00000 -1.00000 -1.00000 2.00000 -1.00000 -1.00000 2.00000 -1.00000 ... ... ..... ....... ......... ........... ............. E18 * 2.00000 -1.20527 -0.54733 1.86494 -1.70043 0.18454 1.47802 -1.96595 0.89148 0.89148 -1.96595 1.47802 0.18454 -1.70043 1.86494 -0.54733 -1.20527 2.00000 -1.20527 -0.54733 1.86494 -1.70043 0.18454 1.47802 -1.96595 0.89148 ... ... ..... ....... ......... ........... ............. E19 * 2.00000 -1.39227 -0.06159 1.47802 -1.99621 1.30124 0.18454 -1.55816 1.98484 -1.20527 -0.30678 1.63239 -1.96595 1.10473 0.42787 -1.70043 1.93959 -1.00000 -0.54733 1.76202 -1.90588 0.89148 0.66471 -1.81693 1.86494 -0.77957 ... ... ..... ....... ......... ........... ............. E20 * 2.00000 -1.55816 0.42787 0.89148 -1.81693 1.93959 -1.20527 -0.06159 1.30124 -1.96595 1.76202 -0.77957 -0.54733 1.63239 -1.99621 1.47802 -0.30678 -1.00000 1.86494 -1.90588 1.10473 0.18454 -1.39227 1.98484 -1.70043 0.66471 ... ... ..... ....... ......... ........... ............. E21 * 2.00000 -1.70043 0.89148 0.18454 -1.20527 1.86494 -1.96595 1.47802 -0.54733 -0.54733 1.47802 -1.96595 1.86494 -1.20527 0.18454 0.89148 -1.70043 2.00000 -1.70043 0.89148 0.18454 -1.20527 1.86494 -1.96595 1.47802 -0.54733 ... ... ..... ....... ......... ........... ............. E22 * 2.00000 -1.81693 1.30124 -0.54733 -0.30678 1.10473 -1.70043 1.98484 -1.90588 1.47802 -0.77957 -0.06159 0.89148 -1.55816 1.93959 -1.96595 1.63239 -1.00000 0.18454 0.66471 -1.39227 1.86494 -1.99621 1.76202 -1.20527 0.42787 ... ... ..... ....... ......... ........... ............. E23 * 2.00000 -1.90588 1.63239 -1.20527 0.66471 -0.06159 -0.54733 1.10473 -1.55816 1.86494 -1.99621 1.93959 -1.70043 1.30124 -0.77957 0.18454 0.42787 -1.00000 1.47802 -1.81693 1.98484 -1.96595 1.76202 -1.39227 0.89148 -0.30678 ... ... ..... ....... ......... ........... ............. E24 * 2.00000 -1.96595 1.86494 -1.70043 1.47802 -1.20527 0.89148 -0.54733 0.18454 0.18454 -0.54733 0.89148 -1.20527 1.47802 -1.70043 1.86494 -1.96595 2.00000 -1.96595 1.86494 -1.70043 1.47802 -1.20527 0.89148 -0.54733 0.18454 ... ... ..... ....... ......... ........... ............. E25 * 2.00000 -1.99621 1.98484 -1.96595 1.93959 -1.90588 1.86494 -1.81693 1.76202 -1.70043 1.63239 -1.55816 1.47802 -1.39227 1.30124 -1.20527 1.10473 -1.00000 0.89148 -0.77957 0.66471 -0.54733 0.42787 -0.30678 0.18454 -0.06159 ... ... ..... ....... ......... ........... ............. Irrational character values: 1.984841019344 = 2*cos(2*π/51) = (1+√17+√34+2*√17−2*√17−3*√17+√34+2*√17−2*√34−2*√17)/16 + √6*√17+√17−√34+2*√17+2*√17−3*√17−√34+2*√17+2*√34−2*√17/8 1.939593872070 = 2*cos(4*π/51) = (1+√17−√34+2*√17+2*√17−3*√17−√34+2*√17+2*√34−2*√17)/16 + √6*√17+√17+√34+2*√17+2*√17−3*√17+√34+2*√17−2*√34−2*√17/8 1.864944458809 = 2*cos(6*π/51) = 2*cos(2*π/17) = (−1+√17+√34−2*√17+2*√17+3*√17−√34−2*√17−2*√34+2*√17)/8 1.762024388572 = 2*cos(8*π/51) = (1+√17+√34+2*√17+2*√17−3*√17+√34+2*√17−2*√34−2*√17)/16 + √6*√17+√17−√34+2*√17−2*√17−3*√17−√34+2*√17+2*√34−2*√17/8 1.632393824712 = 2*cos(10*π/51) = (1−√17−√34−2*√17+2*√17+3*√17−√34−2*√17−2*√34+2*√17)/16 + √6*√17−√17+√34−2*√17+2*√17+3*√17+√34−2*√17+2*√34+2*√17/8 1.478017834441 = 2*cos(12*π/51) = 2*cos(4*π/17) = (−1+√17−√34−2*√17+2*√17+3*√17+√34−2*√17+2*√34+2*√17)/8 1.301236600408 = 2*cos(14*π/51) = (1−√17+√34−2*√17+2*√17+3*√17+√34−2*√17+2*√34+2*√17)/16 + √6*√17−√17−√34−2*√17−2*√17+3*√17−√34−2*√17−2*√34+2*√17/8 1.104729945921 = 2*cos(16*π/51) = (1+√17−√34+2*√17−2*√17−3*√17−√34+2*√17+2*√34−2*√17)/16 + √6*√17+√17+√34+2*√17−2*√17−3*√17+√34+2*√17−2*√34−2*√17/8 0.891476711553 = 2*cos(18*π/51) = 2*cos(6*π/17) = (−1−√17+√34+2*√17+2*√17−3*√17−√34+2*√17+2*√34−2*√17)/8 0.664709598959 = 2*cos(20*π/51) = (1−√17+√34−2*√17+2*√17+3*√17+√34−2*√17+2*√34+2*√17)/16 − √6*√17−√17−√34−2*√17−2*√17+3*√17−√34−2*√17−2*√34+2*√17/8 0.427866166413 = 2*cos(22*π/51) = (1−√17+√34−2*√17−2*√17+3*√17+√34−2*√17+2*√34+2*√17)/16 + √6*√17−√17−√34−2*√17+2*√17+3*√17−√34−2*√17−2*√34+2*√17/8 0.184536718927 = 2*cos(24*π/51) = 2*cos(8*π/17) = (−1+√17+√34−2*√17−2*√17+3*√17−√34−2*√17−2*√34+2*√17)/8 -0.061590117112 = 2*cos(26*π/51) = −((−1−√17−√34+2*√17−2*√17−3*√17+√34+2*√17−2*√34−2*√17)/16 + √6*√17+√17−√34+2*√17−2*√17−3*√17−√34+2*√17+2*√34−2*√17/8) -0.306783309757 = 2*cos(28*π/51) = −((−1+√17+√34−2*√17+2*√17+3*√17−√34−2*√17−2*√34+2*√17)/16 − √6*√17−√17+√34−2*√17−2*√17+3*√17+√34−2*√17+2*√34+2*√17/8) -0.547325980144 = 2*cos(30*π/51) = 2*cos(10*π/17) = −(1+√17−√34+2*√17+2*√17−3*√17−√34+2*√17+2*√34−2*√17)/8 -0.779571746585 = 2*cos(32*π/51) = −((−1−√17−√34+2*√17+2*√17−3*√17+√34+2*√17−2*√34−2*√17)/16 + √6*√17+√17−√34+2*√17+2*√17−3*√17−√34+2*√17+2*√34−2*√17/8) -1.205269272759 = 2*cos(36*π/51) = 2*cos(12*π/17) = −(1+√17+√34+2*√17−2*√17−3*√17+√34+2*√17−2*√34−2*√17)/8 -1.392267891926 = 2*cos(38*π/51) = −((−1−√17+√34+2*√17−2*√17−3*√17−√34+2*√17+2*√34−2*√17)/16 + √6*√17+√17+√34+2*√17+2*√17−3*√17+√34+2*√17−2*√34−2*√17/8) -1.558161149051 = 2*cos(40*π/51) = −((−1+√17+√34−2*√17+2*√17+3*√17−√34−2*√17−2*√34+2*√17)/16 + √6*√17−√17+√34−2*√17−2*√17+3*√17+√34−2*√17+2*√34+2*√17/8) -1.700434271459 = 2*cos(42*π/51) = 2*cos(14*π/17) = −(1+√17+√34+2*√17+2*√17−3*√17+√34+2*√17−2*√34−2*√17)/8 -1.816930543639 = 2*cos(44*π/51) = −((−1+√17+√34−2*√17−2*√17+3*√17−√34−2*√17−2*√34+2*√17)/16 + √6*√17−√17+√34−2*√17+2*√17+3*√17+√34−2*√17+2*√34+2*√17/8) -1.905884000854 = 2*cos(46*π/51) = −((−1+√17−√34−2*√17+2*√17+3*√17+√34−2*√17+2*√34+2*√17)/16 + √6*√17−√17−√34−2*√17+2*√17+3*√17−√34−2*√17−2*√34+2*√17/8) -1.965946199368 = 2*cos(48*π/51) = 2*cos(16*π/17) = −(1−√17+√34−2*√17+2*√17+3*√17+√34−2*√17+2*√34+2*√17)/8 -1.996206657474 = 2*cos(50*π/51) = −((−1−√17+√34+2*√17+2*√17−3*√17−√34+2*√17+2*√34−2*√17)/16 + √6*√17+√17+√34+2*√17−2*√17−3*√17+√34+2*√17−2*√34−2*√17/8) Symmetry of Rotations and Cartesian products A R+p+d+f+g+h+i+j+k+l+m Rz, z, z2, z3, z4, z5, z6 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 C51 point group is 51, and the order of the principal axis (C51) is 51. The group has 26 irreducible representations. β The C51 point group is generated by one single symmetry element, C51. Therefore, it is a cyclic group. γ The lowest nonvanishing multipole moment in C51 is 2 (dipole moment). δ This is an Abelian point group (the commutative law holds between all symmetry operations). The C51 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 50 cases have been combined into 25 two-dimensional representations that are no longer irreducible but have real-valued characters. Accordingly, 25 pairs of left and right rotations have been combined into one two-membered pseudo-class each. ζ The 25 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. 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 That a regular 17-gon can be constructed with compass and ruler was unknown to mathematicians until Gauss proved it in 1796. The first actual construction was performed thirty years later by Johannes Erchinger in 1825. The regular 51-gon can be derived from the regular 17-gon by overlaying it with a equilateral triangle.
| C49 | ||
| C50 | ||
| C51 | C51v C51h D51 D51h D51d | |
| C52 | ||
| C53 |
This Character Table for the C51 point group was created by Gernot Katzer.
For other groups and some explanations, see the Main Page.