Character Table for Point Group D204d

Character table for the D_204d point group

D204d E 2 S408 2 C204 2 S136 2 C102 2 S408^5 2 C68 2 S408^7 2 C51 2 S136^3 2 C204^5 2 S408^11 2 C34 2 S408^13 2 C204^7 2 S136^5 2 C51^2 2 S24 2 C68^3 2 S408^19 2 C102^5 2 S136^7 2 C204^11 2 S408^23 2 C17 2 S408^25 2 C204^13 2 S136^9 2 C102^7 2 S408^29 2 C68^5 2 S408^31 2 C51^4 2 S136^11 2 C12 2 S408^35 2 C34^3 2 S408^37 2 C204^19 2 S136^13 2 C51^5 2 S408^41 2 C68^7 2 S408^43 2 C102^11 2 S136^15 2 C204^23 2 S408^47 2 C17^2 2 S408^49 2 C204^25 2 S8 2 C102^13 2 S408^53 2 C68^9 2 S408^55 2 C51^7 2 S136^19 2 C204^29 2 S408^59 2 C34^5 2 S408^61 2 C204^31 2 S136^21 2 C51^8 2 S408^65 2 C68^11 2 S408^67 2 C6 2 S136^23 2 C204^35 2 S408^71 2 C17^3 2 S408^73 2 C204^37 2 S136^25 2 C102^19 2 S408^77 2 C68^13 2 S408^79 2 C51^10 2 S136^27 2 C204^41 2 S408^83 2 C34^7 2 S24^5 2 C204^43 2 S136^29 2 C51^11 2 S408^89 2 C68^15 2 S408^91 2 C102^23 2 S136^31 2 C204^47 2 S408^95 2 C17^4 2 S408^97 2 C204^49 2 S136^33 2 C102^25 2 S408^101 2 C4 2 S408^103 2 C51^13 2 S136^35 2 C204^53 2 S408^107 2 C34^9 2 S408^109 2 C204^55 2 S136^37 2 C51^14 2 S408^113 2 C68^19 2 S408^115 2 C102^29 2 S136^39 2 C204^59 2 S24^7 2 C17^5 2 S408^121 2 C204^61 2 S136^41 2 C102^31 2 S408^125 2 C68^21 2 S408^127 2 C51^16 2 S136^43 2 C204^65 2 S408^131 2 C34^11 2 S408^133 2 C204^67 2 S136^45 2 C3 2 S408^137 2 C68^23 2 S408^139 2 C102^35 2 S136^47 2 C204^71 2 S408^143 2 C17^6 2 S408^145 2 C204^73 2 S136^49 2 C102^37 2 S408^149 2 C68^25 2 S408^151 2 C51^19 2 S8^3 2 C204^77 2 S408^155 2 C34^13 2 S408^157 2 C204^79 2 S136^53 2 C51^20 2 S408^161 2 C68^27 2 S408^163 2 C102^41 2 S136^55 2 C204^83 2 S408^167 2 C17^7 2 S408^169 2 C12^5 2 S136^57 2 C102^43 2 S408^173 2 C68^29 2 S408^175 2 C51^22 2 S136^59 2 C204^89 2 S408^179 2 C34^15 2 S408^181 2 C204^91 2 S136^61 2 C51^23 2 S408^185 2 C68^31 2 S24^11 2 C102^47 2 S136^63 2 C204^95 2 S408^191 2 C17^8 2 S408^193 2 C204^97 2 S136^65 2 C102^49 2 S408^197 2 C68^33 2 S408^199 2 C51^25 2 S136^67 2 C204^101 2 S408^203 C2 204 C2' 204 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 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 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 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 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 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 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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 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 -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 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 -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 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 -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 -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 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 -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 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 -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 1.99976 1.99905 1.99787 1.99621 1.99407 1.99147 1.98839 1.98484 1.98082 1.97633 1.97137 1.96595 1.96005 1.95370 1.94688 1.93959 1.93185 1.92365 1.91499 1.90588 1.89632 1.88631 1.87585 1.86494 1.85360 1.84181 1.82959 1.81693 1.80384 1.79033 1.77639 1.76202 1.74724 1.73205 1.71645 1.70043 1.68402 1.66720 1.64999 1.63239 1.61441 1.59603 1.57728 1.55816 1.53867 1.51881 1.49859 1.47802 1.45709 1.43582 1.41421 1.39227 1.36999 1.34739 1.32447 1.30124 1.27769 1.25385 1.22970 1.20527 1.18055 1.15555 1.13027 1.10473 1.07893 1.05286 1.02655 1.00000 0.97321 0.94619 0.91894 0.89148 0.86380 0.83592 0.80784 0.77957 0.75112 0.72248 0.69368 0.66471 0.63558 0.60631 0.57688 0.54733 0.51764 0.48783 0.45790 0.42787 0.39773 0.36750 0.33718 0.30678 0.27631 0.24578 0.21518 0.18454 0.15385 0.12312 0.09237 0.06159 0.03080 0.00000 -0.03080 -0.06159 -0.09237 -0.12312 -0.15385 -0.18454 -0.21518 -0.24578 -0.27631 -0.30678 -0.33718 -0.36750 -0.39773 -0.42787 -0.45790 -0.48783 -0.51764 -0.54733 -0.57688 -0.60631 -0.63558 -0.66471 -0.69368 -0.72248 -0.75112 -0.77957 -0.80784 -0.83592 -0.86380 -0.89148 -0.91894 -0.94619 -0.97321 -1.00000 -1.02655 -1.05286 -1.07893 -1.10473 -1.13027 -1.15555 -1.18055 -1.20527 -1.22970 -1.25385 -1.27769 -1.30124 -1.32447 -1.34739 -1.36999 -1.39227 -1.41421 -1.43582 -1.45709 -1.47802 -1.49859 -1.51881 -1.53867 -1.55816 -1.57728 -1.59603 -1.61441 -1.63239 -1.64999 -1.66720 -1.68402 -1.70043 -1.71645 -1.73205 -1.74724 -1.76202 -1.77639 -1.79033 -1.80384 -1.81693 -1.82959 -1.84181 -1.85360 -1.86494 -1.87585 -1.88631 -1.89632 -1.90588 -1.91499 -1.92365 -1.93185 -1.93959 -1.94688 -1.95370 -1.96005 -1.96595 -1.97137 -1.97633 -1.98082 -1.98484 -1.98839 -1.99147 -1.99407 -1.99621 -1.99787 -1.99905 -1.99976 -2.00000 0.00000 0.00000 ... TT. ..... ....TT. ......... ........TT. ............. E2 2.00000 1.99905 1.99621 1.99147 1.98484 1.97633 1.96595 1.95370 1.93959 1.92365 1.90588 1.88631 1.86494 1.84181 1.81693 1.79033 1.76202 1.73205 1.70043 1.66720 1.63239 1.59603 1.55816 1.51881 1.47802 1.43582 1.39227 1.34739 1.30124 1.25385 1.20527 1.15555 1.10473 1.05286 1.00000 0.94619 0.89148 0.83592 0.77957 0.72248 0.66471 0.60631 0.54733 0.48783 0.42787 0.36750 0.30678 0.24578 0.18454 0.12312 0.06159 0.00000 -0.06159 -0.12312 -0.18454 -0.24578 -0.30678 -0.36750 -0.42787 -0.48783 -0.54733 -0.60631 -0.66471 -0.72248 -0.77957 -0.83592 -0.89148 -0.94619 -1.00000 -1.05286 -1.10473 -1.15555 -1.20527 -1.25385 -1.30124 -1.34739 -1.39227 -1.43582 -1.47802 -1.51881 -1.55816 -1.59603 -1.63239 -1.66720 -1.70043 -1.73205 -1.76202 -1.79033 -1.81693 -1.84181 -1.86494 -1.88631 -1.90588 -1.92365 -1.93959 -1.95370 -1.96595 -1.97633 -1.98484 -1.99147 -1.99621 -1.99905 -2.00000 -1.99905 -1.99621 -1.99147 -1.98484 -1.97633 -1.96595 -1.95370 -1.93959 -1.92365 -1.90588 -1.88631 -1.86494 -1.84181 -1.81693 -1.79033 -1.76202 -1.73205 -1.70043 -1.66720 -1.63239 -1.59603 -1.55816 -1.51881 -1.47802 -1.43582 -1.39227 -1.34739 -1.30124 -1.25385 -1.20527 -1.15555 -1.10473 -1.05286 -1.00000 -0.94619 -0.89148 -0.83592 -0.77957 -0.72248 -0.66471 -0.60631 -0.54733 -0.48783 -0.42787 -0.36750 -0.30678 -0.24578 -0.18454 -0.12312 -0.06159 0.00000 0.06159 0.12312 0.18454 0.24578 0.30678 0.36750 0.42787 0.48783 0.54733 0.60631 0.66471 0.72248 0.77957 0.83592 0.89148 0.94619 1.00000 1.05286 1.10473 1.15555 1.20527 1.25385 1.30124 1.34739 1.39227 1.43582 1.47802 1.51881 1.55816 1.59603 1.63239 1.66720 1.70043 1.73205 1.76202 1.79033 1.81693 1.84181 1.86494 1.88631 1.90588 1.92365 1.93959 1.95370 1.96595 1.97633 1.98484 1.99147 1.99621 1.99905 2.00000 0.00000 0.00000 ... ... TT... ....... ....TT... ........... ........TT... E3 2.00000 1.99787 1.99147 1.98082 1.96595 1.94688 1.92365 1.89632 1.86494 1.82959 1.79033 1.74724 1.70043 1.64999 1.59603 1.53867 1.47802 1.41421 1.34739 1.27769 1.20527 1.13027 1.05286 0.97321 0.89148 0.80784 0.72248 0.63558 0.54733 0.45790 0.36750 0.27631 0.18454 0.09237 0.00000 -0.09237 -0.18454 -0.27631 -0.36750 -0.45790 -0.54733 -0.63558 -0.72248 -0.80784 -0.89148 -0.97321 -1.05286 -1.13027 -1.20527 -1.27769 -1.34739 -1.41421 -1.47802 -1.53867 -1.59603 -1.64999 -1.70043 -1.74724 -1.79033 -1.82959 -1.86494 -1.89632 -1.92365 -1.94688 -1.96595 -1.98082 -1.99147 -1.99787 -2.00000 -1.99787 -1.99147 -1.98082 -1.96595 -1.94688 -1.92365 -1.89632 -1.86494 -1.82959 -1.79033 -1.74724 -1.70043 -1.64999 -1.59603 -1.53867 -1.47802 -1.41421 -1.34739 -1.27769 -1.20527 -1.13027 -1.05286 -0.97321 -0.89148 -0.80784 -0.72248 -0.63558 -0.54733 -0.45790 -0.36750 -0.27631 -0.18454 -0.09237 0.00000 0.09237 0.18454 0.27631 0.36750 0.45790 0.54733 0.63558 0.72248 0.80784 0.89148 0.97321 1.05286 1.13027 1.20527 1.27769 1.34739 1.41421 1.47802 1.53867 1.59603 1.64999 1.70043 1.74724 1.79033 1.82959 1.86494 1.89632 1.92365 1.94688 1.96595 1.98082 1.99147 1.99787 2.00000 1.99787 1.99147 1.98082 1.96595 1.94688 1.92365 1.89632 1.86494 1.82959 1.79033 1.74724 1.70043 1.64999 1.59603 1.53867 1.47802 1.41421 1.34739 1.27769 1.20527 1.13027 1.05286 0.97321 0.89148 0.80784 0.72248 0.63558 0.54733 0.45790 0.36750 0.27631 0.18454 0.09237 0.00000 -0.09237 -0.18454 -0.27631 -0.36750 -0.45790 -0.54733 -0.63558 -0.72248 -0.80784 -0.89148 -0.97321 -1.05286 -1.13027 -1.20527 -1.27769 -1.34739 -1.41421 -1.47802 -1.53867 -1.59603 -1.64999 -1.70043 -1.74724 -1.79033 -1.82959 -1.86494 -1.89632 -1.92365 -1.94688 -1.96595 -1.98082 -1.99147 -1.99787 -2.00000 0.00000 0.00000 ... ... ..... TT..... ......... ....TT..... ............. 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TT....... ........... ....TT....... 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Irrational character values: 1.999762845796 = 2*cos(2*π/408) = 2*cos(π/204) 1.999051439427 = 2*cos(4*π/408) = 2*cos(π/102) 1.997865949605 = 2*cos(6*π/408) = 2*cos(π/68) = √8+√2*√17−√17+√34−2*√17+2*√17+3*√17+√34−2*√17+2*√34+2*√17/2 1.996206657474 = 2*cos(8*π/408) = 2*cos(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 1.994073956543 = 2*cos(10*π/408) = 2*cos(5*π/204) 1.991468352590 = 2*cos(12*π/408) = 2*cos(π/34) = √2*√17−√17+√34−2*√17+2*√17+3*√17+√34−2*√17+2*√34+2*√17/4 1.988390463546 = 2*cos(14*π/408) = 2*cos(7*π/204) 1.984841019344 = 2*cos(16*π/408) = 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 1.980820861750 = 2*cos(18*π/408) = 2*cos(3*π/68) = √8+√2*√17+√17+√34+2*√17+2*√17−3*√17+√34+2*√17−2*√34−2*√17/2 1.976330944163 = 2*cos(20*π/408) = 2*cos(5*π/102) 1.971372331383 = 2*cos(22*π/408) = 2*cos(11*π/204) 1.965946199368 = 2*cos(24*π/408) = 2*cos(π/17) = (1−√17+√34−2*√17+2*√17+3*√17+√34−2*√17+2*√34+2*√17)/8 1.960053834947 = 2*cos(26*π/408) = 2*cos(13*π/204) 1.953696635519 = 2*cos(28*π/408) = 2*cos(7*π/102) 1.946876108721 = 2*cos(30*π/408) = 2*cos(5*π/68) = √8+√2*√17+√17+√34+2*√17−2*√17−3*√17+√34+2*√17−2*√34−2*√17/2 1.939593872070 = 2*cos(32*π/408) = 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 1.931851652578 = 2*cos(34*π/408) = 2*cos(π/12) = √2+√3 1.923651286346 = 2*cos(36*π/408) = 2*cos(3*π/34) = √2*√17+√17+√34+2*√17+2*√17−3*√17+√34+2*√17−2*√34−2*√17/4 1.914994718124 = 2*cos(38*π/408) = 2*cos(19*π/204) 1.905884000854 = 2*cos(40*π/408) = 2*cos(5*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 1.896321295182 = 2*cos(42*π/408) = 2*cos(7*π/68) = √8+√2*√17+√17−√34+2*√17+2*√17−3*√17−√34+2*√17+2*√34−2*√17/2 1.886308868943 = 2*cos(44*π/408) = 2*cos(11*π/102) 1.875849096625 = 2*cos(46*π/408) = 2*cos(23*π/204) 1.864944458809 = 2*cos(48*π/408) = 2*cos(2*π/17) = (−1+√17+√34−2*√17+2*√17+3*√17−√34−2*√17−2*√34+2*√17)/8 1.853597541574 = 2*cos(50*π/408) = 2*cos(25*π/204) 1.841811035890 = 2*cos(52*π/408) = 2*cos(13*π/102) 1.829587736976 = 2*cos(54*π/408) = 2*cos(9*π/68) = √8+√2*√17−√17−√34−2*√17+2*√17+3*√17−√34−2*√17−2*√34+2*√17/2 1.816930543639 = 2*cos(56*π/408) = 2*cos(7*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 1.803842457586 = 2*cos(58*π/408) = 2*cos(29*π/204) 1.790326582710 = 2*cos(60*π/408) = 2*cos(5*π/34) = √2*√17+√17+√34+2*√17−2*√17−3*√17+√34+2*√17−2*√34−2*√17/4 1.776386124359 = 2*cos(62*π/408) = 2*cos(31*π/204) 1.762024388572 = 2*cos(64*π/408) = 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 1.747244781293 = 2*cos(66*π/408) = 2*cos(11*π/68) = √8+√2*√17+√17−√34+2*√17−2*√17−3*√17−√34+2*√17+2*√34−2*√17/2 1.732050807569 = 2*cos(68*π/408) = 2*cos(π/6) = √3 1.716446070715 = 2*cos(70*π/408) = 2*cos(35*π/204) 1.700434271459 = 2*cos(72*π/408) = 2*cos(3*π/17) = (1+√17+√34+2*√17+2*√17−3*√17+√34+2*√17−2*√34−2*√17)/8 1.684019207068 = 2*cos(74*π/408) = 2*cos(37*π/204) 1.667204770442 = 2*cos(76*π/408) = 2*cos(19*π/102) 1.649994949197 = 2*cos(78*π/408) = 2*cos(13*π/68) = √8+√2*√17−√17+√34−2*√17−2*√17+3*√17+√34−2*√17+2*√34+2*√17/2 1.632393824712 = 2*cos(80*π/408) = 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 1.614405571170 = 2*cos(82*π/408) = 2*cos(41*π/204) 1.596034454560 = 2*cos(84*π/408) = 2*cos(7*π/34) = √2*√17+√17−√34+2*√17+2*√17−3*√17−√34+2*√17+2*√34−2*√17/4 1.577284831670 = 2*cos(86*π/408) = 2*cos(43*π/204) 1.558161149051 = 2*cos(88*π/408) = 2*cos(11*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 1.538667941966 = 2*cos(90*π/408) = 2*cos(15*π/68) = √8+√2*√17−√17−√34−2*√17−2*√17+3*√17−√34−2*√17−2*√34+2*√17/2 1.518809833309 = 2*cos(92*π/408) = 2*cos(23*π/102) 1.498591532516 = 2*cos(94*π/408) = 2*cos(47*π/204) 1.478017834441 = 2*cos(96*π/408) = 2*cos(4*π/17) = (−1+√17−√34−2*√17+2*√17+3*√17+√34−2*√17+2*√34+2*√17)/8 1.457093618224 = 2*cos(98*π/408) = 2*cos(49*π/204) 1.435823846129 = 2*cos(100*π/408) = 2*cos(25*π/102) 1.414213562373 = 2*cos(102*π/408) = 2*cos(π/4) = √2 1.392267891926 = 2*cos(104*π/408) = 2*cos(13*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 1.369992039295 = 2*cos(106*π/408) = 2*cos(53*π/204) 1.347391287293 = 2*cos(108*π/408) = 2*cos(9*π/34) = √2*√17−√17−√34−2*√17+2*√17+3*√17−√34−2*√17−2*√34+2*√17/4 1.324470995783 = 2*cos(110*π/408) = 2*cos(55*π/204) 1.301236600408 = 2*cos(112*π/408) = 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 1.277693611304 = 2*cos(114*π/408) = 2*cos(19*π/68) = √8−√2*√17−√17−√34−2*√17−2*√17+3*√17−√34−2*√17−2*√34+2*√17/2 1.253847611788 = 2*cos(116*π/408) = 2*cos(29*π/102) 1.229704257040 = 2*cos(118*π/408) = 2*cos(59*π/204) 1.205269272759 = 2*cos(120*π/408) = 2*cos(5*π/17) = (1+√17+√34+2*√17−2*√17−3*√17+√34+2*√17−2*√34−2*√17)/8 1.180548453802 = 2*cos(122*π/408) = 2*cos(61*π/204) 1.155547662817 = 2*cos(124*π/408) = 2*cos(31*π/102) 1.130272828845 = 2*cos(126*π/408) = 2*cos(21*π/68) = √8−√2*√17−√17+√34−2*√17−2*√17+3*√17+√34−2*√17+2*√34+2*√17/2 1.104729945921 = 2*cos(128*π/408) = 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 1.078925071646 = 2*cos(130*π/408) = 2*cos(65*π/204) 1.052864325755 = 2*cos(132*π/408) = 2*cos(11*π/34) = √2*√17+√17−√34+2*√17−2*√17−3*√17−√34+2*√17+2*√34−2*√17/4 1.026553888662 = 2*cos(134*π/408) = 2*cos(67*π/204) 0.973208957134 = 2*cos(138*π/408) = 2*cos(23*π/68) = √8−√2*√17+√17−√34+2*√17−2*√17−3*√17−√34+2*√17+2*√34−2*√17/2 0.946187113672 = 2*cos(140*π/408) = 2*cos(35*π/102) 0.918940877959 = 2*cos(142*π/408) = 2*cos(71*π/204) 0.891476711553 = 2*cos(144*π/408) = 2*cos(6*π/17) = (−1−√17+√34+2*√17+2*√17−3*√17−√34+2*√17+2*√34−2*√17)/8 0.863801127698 = 2*cos(146*π/408) = 2*cos(73*π/204) 0.835920689774 = 2*cos(148*π/408) = 2*cos(37*π/102) 0.807842009744 = 2*cos(150*π/408) = 2*cos(25*π/68) = √8−√2*√17−√17−√34−2*√17+2*√17+3*√17−√34−2*√17−2*√34+2*√17/2 0.779571746585 = 2*cos(152*π/408) = 2*cos(19*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 0.751116604710 = 2*cos(154*π/408) = 2*cos(77*π/204) 0.722483332374 = 2*cos(156*π/408) = 2*cos(13*π/34) = √2*√17−√17+√34−2*√17−2*√17+3*√17+√34−2*√17+2*√34+2*√17/4 0.693678720079 = 2*cos(158*π/408) = 2*cos(79*π/204) 0.664709598959 = 2*cos(160*π/408) = 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 0.635582839164 = 2*cos(162*π/408) = 2*cos(27*π/68) = √8−√2*√17+√17−√34+2*√17+2*√17−3*√17−√34+2*√17+2*√34−2*√17/2 0.606305348226 = 2*cos(164*π/408) = 2*cos(41*π/102) 0.576884069426 = 2*cos(166*π/408) = 2*cos(83*π/204) 0.547325980144 = 2*cos(168*π/408) = 2*cos(7*π/17) = (1+√17−√34+2*√17+2*√17−3*√17−√34+2*√17+2*√34−2*√17)/8 0.517638090205 = 2*cos(170*π/408) = 2*cos(5*π/12) = √2−√3 0.487827440217 = 2*cos(172*π/408) = 2*cos(43*π/102) 0.457901099900 = 2*cos(174*π/408) = 2*cos(29*π/68) = √8−√2*√17+√17+√34+2*√17−2*√17−3*√17+√34+2*√17−2*√34−2*√17/2 0.427866166413 = 2*cos(176*π/408) = 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 0.397729762666 = 2*cos(178*π/408) = 2*cos(89*π/204) 0.367499035633 = 2*cos(180*π/408) = 2*cos(15*π/34) = √2*√17−√17−√34−2*√17−2*√17+3*√17−√34−2*√17−2*√34+2*√17/4 0.337181154659 = 2*cos(182*π/408) = 2*cos(91*π/204) 0.306783309757 = 2*cos(184*π/408) = 2*cos(23*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 0.276312709904 = 2*cos(186*π/408) = 2*cos(31*π/68) = √8−√2*√17+√17+√34+2*√17+2*√17−3*√17+√34+2*√17−2*√34−2*√17/2 0.245776581329 = 2*cos(188*π/408) = 2*cos(47*π/102) 0.215182165806 = 2*cos(190*π/408) = 2*cos(95*π/204) 0.184536718927 = 2*cos(192*π/408) = 2*cos(8*π/17) = (−1+√17+√34−2*√17−2*√17+3*√17−√34−2*√17−2*√34+2*√17)/8 0.153847508389 = 2*cos(194*π/408) = 2*cos(97*π/204) 0.123121812268 = 2*cos(196*π/408) = 2*cos(49*π/102) 0.092366917291 = 2*cos(198*π/408) = 2*cos(33*π/68) = √8−√2*√17−√17+√34−2*√17+2*√17+3*√17+√34−2*√17+2*√34+2*√17/2 0.061590117112 = 2*cos(200*π/408) = 2*cos(25*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 0.030798710578 = 2*cos(202*π/408) = 2*cos(101*π/204) 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 E195 m E196 l E197 k+m E198 j+l E199 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)} E200 h+j+l {z((x²−y²)²−4x²y²), xyz(x²−y²)} E201 g+i+k+m {xz(x²−3y²), yz(3x²−y²)}, {xz³(x²−3y²), yz³(3x²−y²)} E202 f+h+j+l {z(x²−y²), xyz}, {z³(x²−y²), xyz³} E203 R+d+g+i+k+m {R_x, R_y}, {xz, yz}, {xz³, yz³}, {xz⁵, yz⁵} Notes: α The order of the D_204d point group is 816, and the order of the principal axis (S₄₀₈) is 408. The group has 207 irreducible representations. β The D_204d point group is isomorphic to C_408v and D₄₀₈. γ The D_204d point group is generated by two symmetry elements, S₄₀₈ and either a perpendicular C₂^′ or a vertical σ_d. Also, the group may be generated from a C₂^′ plus a σ_d (some pairs will yield smaller groups, though; choosing a minimum angle is safe). δ The lowest nonvanishing multipole moment in D_204d is 4 (quadrupole moment). ε This point group is non-Abelian (some symmetry operations are not commutative). Therefore, the character table contains multi-membered classes and degenerate irreducible representations. ζ 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. η 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. Some of these algebraic expressions, however, are too complicated to be included in the list above. θ 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. Further halving of angles yields regular polygons of order 102,204,408 etc.

This Character Table for the D_204d point group was created by Gernot Katzer.

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

	D_170d
	D_192d
C₂₀₄ C_204v C_204h D₂₀₄ D_204h	D_204d	S₂₀₄
	D_240d
	D_255d

Character table for the D204d point group

Character table for the D_204d point group