C64v E 2 C64 2 C32 2 C64^3 2 C16 2 C64^5 2 C32^3 2 C64^7 2 C8 2 C64^9 2 C32^5 2 C64^11 2 C16^3 2 C64^13 2 C32^7 2 C64^15 2 C4 2 C64^17 2 C32^9 2 C64^19 2 C16^5 2 C64^21 2 C32^11 2 C64^23 2 C8^3 2 C64^25 2 C32^13 2 C64^27 2 C16^7 2 C64^29 2 C32^15 2 C64^31 C2 32 sv 32 sd <R> <p> <—d—> <——f——> <———g———> <————h————> <—————i—————> A1 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 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 ... ..T ....T ......T ........T ..........T ............T A2 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 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 -1.00000 -1.00000 ..T ... ..... ....... ......... ........... ............. B1 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 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 1.00000 -1.00000 ... ... ..... ....... ......... ........... ............. B2 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 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 ... ... ..... ....... ......... ........... ............. E1 2.00000 1.99037 1.96157 1.91388 1.84776 1.76384 1.66294 1.54602 1.41421 1.26879 1.11114 0.94279 0.76537 0.58057 0.39018 0.19603 0.00000 -0.19603 -0.39018 -0.58057 -0.76537 -0.94279 -1.11114 -1.26879 -1.41421 -1.54602 -1.66294 -1.76384 -1.84776 -1.91388 -1.96157 -1.99037 -2.00000 0.00000 0.00000 TT. TT. ..TT. ....TT. ......TT. ........TT. ..........TT. E2 2.00000 1.96157 1.84776 1.66294 1.41421 1.11114 0.76537 0.39018 0.00000 -0.39018 -0.76537 -1.11114 -1.41421 -1.66294 -1.84776 -1.96157 -2.00000 -1.96157 -1.84776 -1.66294 -1.41421 -1.11114 -0.76537 -0.39018 0.00000 0.39018 0.76537 1.11114 1.41421 1.66294 1.84776 1.96157 2.00000 0.00000 0.00000 ... ... TT... ..TT... ....TT... ......TT... ........TT... E3 2.00000 1.91388 1.66294 1.26879 0.76537 0.19603 -0.39018 -0.94279 -1.41421 -1.76384 -1.96157 -1.99037 -1.84776 -1.54602 -1.11114 -0.58057 0.00000 0.58057 1.11114 1.54602 1.84776 1.99037 1.96157 1.76384 1.41421 0.94279 0.39018 -0.19603 -0.76537 -1.26879 -1.66294 -1.91388 -2.00000 0.00000 0.00000 ... ... ..... TT..... ..TT..... ....TT..... ......TT..... E4 2.00000 1.84776 1.41421 0.76537 0.00000 -0.76537 -1.41421 -1.84776 -2.00000 -1.84776 -1.41421 -0.76537 0.00000 0.76537 1.41421 1.84776 2.00000 1.84776 1.41421 0.76537 0.00000 -0.76537 -1.41421 -1.84776 -2.00000 -1.84776 -1.41421 -0.76537 0.00000 0.76537 1.41421 1.84776 2.00000 0.00000 0.00000 ... ... ..... ....... TT....... ..TT....... ....TT....... E5 2.00000 1.76384 1.11114 0.19603 -0.76537 -1.54602 -1.96157 -1.91388 -1.41421 -0.58057 0.39018 1.26879 1.84776 1.99037 1.66294 0.94279 0.00000 -0.94279 -1.66294 -1.99037 -1.84776 -1.26879 -0.39018 0.58057 1.41421 1.91388 1.96157 1.54602 0.76537 -0.19603 -1.11114 -1.76384 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... TT......... ..TT......... E6 2.00000 1.66294 0.76537 -0.39018 -1.41421 -1.96157 -1.84776 -1.11114 0.00000 1.11114 1.84776 1.96157 1.41421 0.39018 -0.76537 -1.66294 -2.00000 -1.66294 -0.76537 0.39018 1.41421 1.96157 1.84776 1.11114 0.00000 -1.11114 -1.84776 -1.96157 -1.41421 -0.39018 0.76537 1.66294 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... TT........... E7 2.00000 1.54602 0.39018 -0.94279 -1.84776 -1.91388 -1.11114 0.19603 1.41421 1.99037 1.66294 0.58057 -0.76537 -1.76384 -1.96157 -1.26879 0.00000 1.26879 1.96157 1.76384 0.76537 -0.58057 -1.66294 -1.99037 -1.41421 -0.19603 1.11114 1.91388 1.84776 0.94279 -0.39018 -1.54602 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E8 2.00000 1.41421 0.00000 -1.41421 -2.00000 -1.41421 0.00000 1.41421 2.00000 1.41421 0.00000 -1.41421 -2.00000 -1.41421 0.00000 1.41421 2.00000 1.41421 0.00000 -1.41421 -2.00000 -1.41421 0.00000 1.41421 2.00000 1.41421 0.00000 -1.41421 -2.00000 -1.41421 0.00000 1.41421 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E9 2.00000 1.26879 -0.39018 -1.76384 -1.84776 -0.58057 1.11114 1.99037 1.41421 -0.19603 -1.66294 -1.91388 -0.76537 0.94279 1.96157 1.54602 0.00000 -1.54602 -1.96157 -0.94279 0.76537 1.91388 1.66294 0.19603 -1.41421 -1.99037 -1.11114 0.58057 1.84776 1.76384 0.39018 -1.26879 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E10 2.00000 1.11114 -0.76537 -1.96157 -1.41421 0.39018 1.84776 1.66294 0.00000 -1.66294 -1.84776 -0.39018 1.41421 1.96157 0.76537 -1.11114 -2.00000 -1.11114 0.76537 1.96157 1.41421 -0.39018 -1.84776 -1.66294 0.00000 1.66294 1.84776 0.39018 -1.41421 -1.96157 -0.76537 1.11114 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E11 2.00000 0.94279 -1.11114 -1.99037 -0.76537 1.26879 1.96157 0.58057 -1.41421 -1.91388 -0.39018 1.54602 1.84776 0.19603 -1.66294 -1.76384 0.00000 1.76384 1.66294 -0.19603 -1.84776 -1.54602 0.39018 1.91388 1.41421 -0.58057 -1.96157 -1.26879 0.76537 1.99037 1.11114 -0.94279 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E12 2.00000 0.76537 -1.41421 -1.84776 0.00000 1.84776 1.41421 -0.76537 -2.00000 -0.76537 1.41421 1.84776 0.00000 -1.84776 -1.41421 0.76537 2.00000 0.76537 -1.41421 -1.84776 0.00000 1.84776 1.41421 -0.76537 -2.00000 -0.76537 1.41421 1.84776 0.00000 -1.84776 -1.41421 0.76537 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E13 2.00000 0.58057 -1.66294 -1.54602 0.76537 1.99037 0.39018 -1.76384 -1.41421 0.94279 1.96157 0.19603 -1.84776 -1.26879 1.11114 1.91388 0.00000 -1.91388 -1.11114 1.26879 1.84776 -0.19603 -1.96157 -0.94279 1.41421 1.76384 -0.39018 -1.99037 -0.76537 1.54602 1.66294 -0.58057 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E14 2.00000 0.39018 -1.84776 -1.11114 1.41421 1.66294 -0.76537 -1.96157 0.00000 1.96157 0.76537 -1.66294 -1.41421 1.11114 1.84776 -0.39018 -2.00000 -0.39018 1.84776 1.11114 -1.41421 -1.66294 0.76537 1.96157 0.00000 -1.96157 -0.76537 1.66294 1.41421 -1.11114 -1.84776 0.39018 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E15 2.00000 0.19603 -1.96157 -0.58057 1.84776 0.94279 -1.66294 -1.26879 1.41421 1.54602 -1.11114 -1.76384 0.76537 1.91388 -0.39018 -1.99037 0.00000 1.99037 0.39018 -1.91388 -0.76537 1.76384 1.11114 -1.54602 -1.41421 1.26879 1.66294 -0.94279 -1.84776 0.58057 1.96157 -0.19603 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E16 2.00000 0.00000 -2.00000 0.00000 2.00000 0.00000 -2.00000 0.00000 2.00000 0.00000 -2.00000 0.00000 2.00000 0.00000 -2.00000 0.00000 2.00000 0.00000 -2.00000 0.00000 2.00000 0.00000 -2.00000 0.00000 2.00000 0.00000 -2.00000 0.00000 2.00000 0.00000 -2.00000 0.00000 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E17 2.00000 -0.19603 -1.96157 0.58057 1.84776 -0.94279 -1.66294 1.26879 1.41421 -1.54602 -1.11114 1.76384 0.76537 -1.91388 -0.39018 1.99037 0.00000 -1.99037 0.39018 1.91388 -0.76537 -1.76384 1.11114 1.54602 -1.41421 -1.26879 1.66294 0.94279 -1.84776 -0.58057 1.96157 0.19603 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E18 2.00000 -0.39018 -1.84776 1.11114 1.41421 -1.66294 -0.76537 1.96157 0.00000 -1.96157 0.76537 1.66294 -1.41421 -1.11114 1.84776 0.39018 -2.00000 0.39018 1.84776 -1.11114 -1.41421 1.66294 0.76537 -1.96157 0.00000 1.96157 -0.76537 -1.66294 1.41421 1.11114 -1.84776 -0.39018 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E19 2.00000 -0.58057 -1.66294 1.54602 0.76537 -1.99037 0.39018 1.76384 -1.41421 -0.94279 1.96157 -0.19603 -1.84776 1.26879 1.11114 -1.91388 0.00000 1.91388 -1.11114 -1.26879 1.84776 0.19603 -1.96157 0.94279 1.41421 -1.76384 -0.39018 1.99037 -0.76537 -1.54602 1.66294 0.58057 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E20 2.00000 -0.76537 -1.41421 1.84776 0.00000 -1.84776 1.41421 0.76537 -2.00000 0.76537 1.41421 -1.84776 0.00000 1.84776 -1.41421 -0.76537 2.00000 -0.76537 -1.41421 1.84776 0.00000 -1.84776 1.41421 0.76537 -2.00000 0.76537 1.41421 -1.84776 0.00000 1.84776 -1.41421 -0.76537 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E21 2.00000 -0.94279 -1.11114 1.99037 -0.76537 -1.26879 1.96157 -0.58057 -1.41421 1.91388 -0.39018 -1.54602 1.84776 -0.19603 -1.66294 1.76384 0.00000 -1.76384 1.66294 0.19603 -1.84776 1.54602 0.39018 -1.91388 1.41421 0.58057 -1.96157 1.26879 0.76537 -1.99037 1.11114 0.94279 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E22 2.00000 -1.11114 -0.76537 1.96157 -1.41421 -0.39018 1.84776 -1.66294 0.00000 1.66294 -1.84776 0.39018 1.41421 -1.96157 0.76537 1.11114 -2.00000 1.11114 0.76537 -1.96157 1.41421 0.39018 -1.84776 1.66294 0.00000 -1.66294 1.84776 -0.39018 -1.41421 1.96157 -0.76537 -1.11114 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E23 2.00000 -1.26879 -0.39018 1.76384 -1.84776 0.58057 1.11114 -1.99037 1.41421 0.19603 -1.66294 1.91388 -0.76537 -0.94279 1.96157 -1.54602 0.00000 1.54602 -1.96157 0.94279 0.76537 -1.91388 1.66294 -0.19603 -1.41421 1.99037 -1.11114 -0.58057 1.84776 -1.76384 0.39018 1.26879 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E24 2.00000 -1.41421 0.00000 1.41421 -2.00000 1.41421 0.00000 -1.41421 2.00000 -1.41421 0.00000 1.41421 -2.00000 1.41421 0.00000 -1.41421 2.00000 -1.41421 0.00000 1.41421 -2.00000 1.41421 0.00000 -1.41421 2.00000 -1.41421 0.00000 1.41421 -2.00000 1.41421 0.00000 -1.41421 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E25 2.00000 -1.54602 0.39018 0.94279 -1.84776 1.91388 -1.11114 -0.19603 1.41421 -1.99037 1.66294 -0.58057 -0.76537 1.76384 -1.96157 1.26879 0.00000 -1.26879 1.96157 -1.76384 0.76537 0.58057 -1.66294 1.99037 -1.41421 0.19603 1.11114 -1.91388 1.84776 -0.94279 -0.39018 1.54602 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E26 2.00000 -1.66294 0.76537 0.39018 -1.41421 1.96157 -1.84776 1.11114 0.00000 -1.11114 1.84776 -1.96157 1.41421 -0.39018 -0.76537 1.66294 -2.00000 1.66294 -0.76537 -0.39018 1.41421 -1.96157 1.84776 -1.11114 0.00000 1.11114 -1.84776 1.96157 -1.41421 0.39018 0.76537 -1.66294 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E27 2.00000 -1.76384 1.11114 -0.19603 -0.76537 1.54602 -1.96157 1.91388 -1.41421 0.58057 0.39018 -1.26879 1.84776 -1.99037 1.66294 -0.94279 0.00000 0.94279 -1.66294 1.99037 -1.84776 1.26879 -0.39018 -0.58057 1.41421 -1.91388 1.96157 -1.54602 0.76537 0.19603 -1.11114 1.76384 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E28 2.00000 -1.84776 1.41421 -0.76537 0.00000 0.76537 -1.41421 1.84776 -2.00000 1.84776 -1.41421 0.76537 0.00000 -0.76537 1.41421 -1.84776 2.00000 -1.84776 1.41421 -0.76537 0.00000 0.76537 -1.41421 1.84776 -2.00000 1.84776 -1.41421 0.76537 0.00000 -0.76537 1.41421 -1.84776 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E29 2.00000 -1.91388 1.66294 -1.26879 0.76537 -0.19603 -0.39018 0.94279 -1.41421 1.76384 -1.96157 1.99037 -1.84776 1.54602 -1.11114 0.58057 0.00000 -0.58057 1.11114 -1.54602 1.84776 -1.99037 1.96157 -1.76384 1.41421 -0.94279 0.39018 0.19603 -0.76537 1.26879 -1.66294 1.91388 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E30 2.00000 -1.96157 1.84776 -1.66294 1.41421 -1.11114 0.76537 -0.39018 0.00000 0.39018 -0.76537 1.11114 -1.41421 1.66294 -1.84776 1.96157 -2.00000 1.96157 -1.84776 1.66294 -1.41421 1.11114 -0.76537 0.39018 0.00000 -0.39018 0.76537 -1.11114 1.41421 -1.66294 1.84776 -1.96157 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E31 2.00000 -1.99037 1.96157 -1.91388 1.84776 -1.76384 1.66294 -1.54602 1.41421 -1.26879 1.11114 -0.94279 0.76537 -0.58057 0.39018 -0.19603 0.00000 0.19603 -0.39018 0.58057 -0.76537 0.94279 -1.11114 1.26879 -1.41421 1.54602 -1.66294 1.76384 -1.84776 1.91388 -1.96157 1.99037 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. Irrational character values: 1.990369453344 = 2*cos(2*π/64) = 2*cos(π/32) = √2+√2+√2+√2 1.961570560806 = 2*cos(4*π/64) = 2*cos(π/16) = √2+√2+√2 1.913880671464 = 2*cos(6*π/64) = 2*cos(3*π/32) = √2+√2+√2−√2 1.847759065023 = 2*cos(8*π/64) = 2*cos(π/8) = √2+√2 1.763842528697 = 2*cos(10*π/64) = 2*cos(5*π/32) = √2+√2−√2−√2 1.662939224605 = 2*cos(12*π/64) = 2*cos(3*π/16) = √2+√2−√2 1.546020906725 = 2*cos(14*π/64) = 2*cos(7*π/32) = √2+√2−√2+√2 1.414213562373 = 2*cos(16*π/64) = 2*cos(π/4) = √2 1.268786568327 = 2*cos(18*π/64) = 2*cos(9*π/32) = √2−√2−√2+√2 1.111140466039 = 2*cos(20*π/64) = 2*cos(5*π/16) = √2−√2−√2 0.942793473652 = 2*cos(22*π/64) = 2*cos(11*π/32) = √2−√2−√2−√2 0.765366864730 = 2*cos(24*π/64) = 2*cos(3*π/8) = √2−√2 0.580569354509 = 2*cos(26*π/64) = 2*cos(13*π/32) = √2−√2+√2−√2 0.390180644032 = 2*cos(28*π/64) = 2*cos(7*π/16) = √2−√2+√2 0.196034280659 = 2*cos(30*π/64) = 2*cos(15*π/32) = √2−√2+√2+√2 Symmetry of Rotations and Cartesian products A1 p+d+f+g+h+i+j+k+l+m z, z2, z3, z4, z5, z6 A2 R Rz 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 C64v point group is 128, and the order of the principal axis (C64) is 64. The group has 35 irreducible representations. β The C64v point group is isomorphic to D32d and D64. γ The C64v point group is generated by two symmetry elements, C64 and any σv (or, non-canonically, any σd). Also, the group may be generated from any σv plus any σd planes. δ There are two different sets of symmetry planes containing the principal axis (z axis in standard orientation). By convention, the set denoted as σv contains both the xz and the yz planes. ε The lowest nonvanishing multipole moment in C64v is 2 (dipole 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 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 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.
| C62v | ||
| C63v | ||
| C64 | C64v | C64h D64 D64h D64d S64 |
| C65v | ||
| C66v |
This Character Table for the C64v point group was created by Gernot Katzer.
For other groups and some explanations, see the Main Page.