Character Table for Point Group D32d

Character table for the D_32d point group

D32d E 2 S64 2 C32 2 S64^3 2 C16 2 S64^5 2 C32^3 2 S64^7 2 C8 2 S64^9 2 C32^5 2 S64^11 2 C16^3 2 S64^13 2 C32^7 2 S64^15 2 C4 2 S64^17 2 C32^9 2 S64^19 2 C16^5 2 S64^21 2 C32^11 2 S64^23 2 C8^3 2 S64^25 2 C32^13 2 S64^27 2 C16^7 2 S64^29 2 C32^15 2 S64^31 C2 32 C2' 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 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 ... ..T ..... ......T ......... ..........T ............. 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. ............. 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... 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..... ............. 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....... 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......... ............. 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 ... ... ..... ....... ......... ........... ..TT......... 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 ... ... ..... ....... ......... ..TT....... ............. 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 ... ... ..... ....... ..TT..... ........... ......TT..... 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 ... ... ..... ..TT... ......... ......TT... ............. 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 TT. ... ..TT. ....... ......TT. ........... ..........TT. 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 d+g+i+k+m z², z⁴, z⁶ A2 R R_z B2 p+f+h+j+l z, z³, z⁵ E1 p+f+h+j+l {x, y}, {xz², yz²}, {xz⁴, yz⁴} E2 d+g+i+k+m {x²−y², xy}, {z²(x²−y²), xyz²}, {z⁴(x²−y²), xyz⁴} E3 f+h+j+l {x(x²−3y²), y(3x²−y²)}, {xz²(x²−3y²), yz²(3x²−y²)} E4 g+i+k+m {(x²−y²)²−4x²y², xy(x²−y²)}, {z²((x²−y²)²−4x²y²), xyz²(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)} E6 i+k+m {x²(x²−3y²)²−y²(3x²−y²)², xy(x²−3y²)(3x²−y²)} E7 j+l E8 k+m E9 l E10 m E23 m E24 l E25 k+m E26 j+l E27 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)} E28 h+j+l {z((x²−y²)²−4x²y²), xyz(x²−y²)} E29 g+i+k+m {xz(x²−3y²), yz(3x²−y²)}, {xz³(x²−3y²), yz³(3x²−y²)} E30 f+h+j+l {z(x²−y²), xyz}, {z³(x²−y²), xyz³} E31 R+d+g+i+k+m {R_x, R_y}, {xz, yz}, {xz³, yz³}, {xz⁵, yz⁵} Notes: α The order of the D_32d point group is 128, and the order of the principal axis (S₆₄) is 64. The group has 35 irreducible representations. β The D_32d point group is isomorphic to C_64v and D₆₄. γ The D_32d point group is generated by two symmetry elements, S₆₄ and either a perpendicular C₂^′ or a vertical σ_d. Also, the group may be generated from any C₂^′ plus any σ_d plane. δ The group contains one set of twofold symmetry axes (C₂^′) perpendicular to the principal (z) axis. Both x and y axes are members of that set. ε The single σ_d set of symmetry planes contains neither the xz nor the yz planes. ζ The lowest nonvanishing multipole moment in D_32d is 4 (quadrupole moment). η This point group is non-Abelian (some symmetry operations are not commutative). Therefore, the character table contains multi-membered classes and degenerate irreducible representations. 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.

This Character Table for the D_32d point group was created by Gernot Katzer.

For other groups and some explanations, see the Main Page.

	D_30d
	D_31d
C₃₂ C_32v C_32h D₃₂ D_32h	D_32d	S₃₂
	D_33d
	D_34d

Character table for the D32d point group

Character table for the D_32d point group