D136d E 2 S272 2 C136 2 S272^3 2 C68 2 S272^5 2 C136^3 2 S272^7 2 C34 2 S272^9 2 C136^5 2 S272^11 2 C68^3 2 S272^13 2 C136^7 2 S272^15 2 C17 2 S16 2 C136^9 2 S272^19 2 C68^5 2 S272^21 2 C136^11 2 S272^23 2 C34^3 2 S272^25 2 C136^13 2 S272^27 2 C68^7 2 S272^29 2 C136^15 2 S272^31 2 C17^2 2 S272^33 2 C8 2 S272^35 2 C68^9 2 S272^37 2 C136^19 2 S272^39 2 C34^5 2 S272^41 2 C136^21 2 S272^43 2 C68^11 2 S272^45 2 C136^23 2 S272^47 2 C17^3 2 S272^49 2 C136^25 2 S16^3 2 C68^13 2 S272^53 2 C136^27 2 S272^55 2 C34^7 2 S272^57 2 C136^29 2 S272^59 2 C68^15 2 S272^61 2 C136^31 2 S272^63 2 C17^4 2 S272^65 2 C136^33 2 S272^67 2 C4 2 S272^69 2 C136^35 2 S272^71 2 C34^9 2 S272^73 2 C136^37 2 S272^75 2 C68^19 2 S272^77 2 C136^39 2 S272^79 2 C17^5 2 S272^81 2 C136^41 2 S272^83 2 C68^21 2 S16^5 2 C136^43 2 S272^87 2 C34^11 2 S272^89 2 C136^45 2 S272^91 2 C68^23 2 S272^93 2 C136^47 2 S272^95 2 C17^6 2 S272^97 2 C136^49 2 S272^99 2 C68^25 2 S272^101 2 C8^3 2 S272^103 2 C34^13 2 S272^105 2 C136^53 2 S272^107 2 C68^27 2 S272^109 2 C136^55 2 S272^111 2 C17^7 2 S272^113 2 C136^57 2 S272^115 2 C68^29 2 S272^117 2 C136^59 2 S16^7 2 C34^15 2 S272^121 2 C136^61 2 S272^123 2 C68^31 2 S272^125 2 C136^63 2 S272^127 2 C17^8 2 S272^129 2 C136^65 2 S272^131 2 C68^33 2 S272^133 2 C136^67 2 S272^135 C2 136 C2' 136 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 ... ... ....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 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 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 ... ... ..... ....... ......... ........... ............. 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 ... ..T ..... ......T ......... ..........T ............. 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TT....... ........... ....TT....... 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TT........... 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E130 2.00000 -1.98082 1.92365 -1.82959 1.70043 -1.53867 1.34739 -1.13027 0.89148 -0.63558 0.36750 -0.09237 -0.18454 0.45790 -0.72248 0.97321 -1.20527 1.41421 -1.59603 1.74724 -1.86494 1.94688 -1.99147 1.99787 -1.96595 1.89632 -1.79033 1.64999 -1.47802 1.27769 -1.05286 0.80784 -0.54733 0.27631 0.00000 -0.27631 0.54733 -0.80784 1.05286 -1.27769 1.47802 -1.64999 1.79033 -1.89632 1.96595 -1.99787 1.99147 -1.94688 1.86494 -1.74724 1.59603 -1.41421 1.20527 -0.97321 0.72248 -0.45790 0.18454 0.09237 -0.36750 0.63558 -0.89148 1.13027 -1.34739 1.53867 -1.70043 1.82959 -1.92365 1.98082 -2.00000 1.98082 -1.92365 1.82959 -1.70043 1.53867 -1.34739 1.13027 -0.89148 0.63558 -0.36750 0.09237 0.18454 -0.45790 0.72248 -0.97321 1.20527 -1.41421 1.59603 -1.74724 1.86494 -1.94688 1.99147 -1.99787 1.96595 -1.89632 1.79033 -1.64999 1.47802 -1.27769 1.05286 -0.80784 0.54733 -0.27631 0.00000 0.27631 -0.54733 0.80784 -1.05286 1.27769 -1.47802 1.64999 -1.79033 1.89632 -1.96595 1.99787 -1.99147 1.94688 -1.86494 1.74724 -1.59603 1.41421 -1.20527 0.97321 -0.72248 0.45790 -0.18454 -0.09237 0.36750 -0.63558 0.89148 -1.13027 1.34739 -1.53867 1.70043 -1.82959 1.92365 -1.98082 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. 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E132 2.00000 -1.99147 1.96595 -1.92365 1.86494 -1.79033 1.70043 -1.59603 1.47802 -1.34739 1.20527 -1.05286 0.89148 -0.72248 0.54733 -0.36750 0.18454 0.00000 -0.18454 0.36750 -0.54733 0.72248 -0.89148 1.05286 -1.20527 1.34739 -1.47802 1.59603 -1.70043 1.79033 -1.86494 1.92365 -1.96595 1.99147 -2.00000 1.99147 -1.96595 1.92365 -1.86494 1.79033 -1.70043 1.59603 -1.47802 1.34739 -1.20527 1.05286 -0.89148 0.72248 -0.54733 0.36750 -0.18454 0.00000 0.18454 -0.36750 0.54733 -0.72248 0.89148 -1.05286 1.20527 -1.34739 1.47802 -1.59603 1.70043 -1.79033 1.86494 -1.92365 1.96595 -1.99147 2.00000 -1.99147 1.96595 -1.92365 1.86494 -1.79033 1.70043 -1.59603 1.47802 -1.34739 1.20527 -1.05286 0.89148 -0.72248 0.54733 -0.36750 0.18454 0.00000 -0.18454 0.36750 -0.54733 0.72248 -0.89148 1.05286 -1.20527 1.34739 -1.47802 1.59603 -1.70043 1.79033 -1.86494 1.92365 -1.96595 1.99147 -2.00000 1.99147 -1.96595 1.92365 -1.86494 1.79033 -1.70043 1.59603 -1.47802 1.34739 -1.20527 1.05286 -0.89148 0.72248 -0.54733 0.36750 -0.18454 0.00000 0.18454 -0.36750 0.54733 -0.72248 0.89148 -1.05286 1.20527 -1.34739 1.47802 -1.59603 1.70043 -1.79033 1.86494 -1.92365 1.96595 -1.99147 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ..TT....... ............. E133 2.00000 -1.99520 1.98082 -1.95693 1.92365 -1.88113 1.82959 -1.76926 1.70043 -1.62345 1.53867 -1.44650 1.34739 -1.24181 1.13027 -1.01331 0.89148 -0.76537 0.63558 -0.50275 0.36750 -0.23049 0.09237 0.04620 -0.18454 0.32199 -0.45790 0.59161 -0.72248 0.84989 -0.97321 1.09186 -1.20527 1.31289 -1.41421 1.50875 -1.59603 1.67566 -1.74724 1.81044 -1.86494 1.91050 -1.94688 1.97391 -1.99147 1.99947 -1.99787 1.98667 -1.96595 1.93578 -1.89632 1.84776 -1.79033 1.72430 -1.64999 1.56777 -1.47802 1.38117 -1.27769 1.16808 -1.05286 0.93259 -0.80784 0.67921 -0.54733 0.41281 -0.27631 0.13849 0.00000 -0.13849 0.27631 -0.41281 0.54733 -0.67921 0.80784 -0.93259 1.05286 -1.16808 1.27769 -1.38117 1.47802 -1.56777 1.64999 -1.72430 1.79033 -1.84776 1.89632 -1.93578 1.96595 -1.98667 1.99787 -1.99947 1.99147 -1.97391 1.94688 -1.91050 1.86494 -1.81044 1.74724 -1.67566 1.59603 -1.50875 1.41421 -1.31289 1.20527 -1.09186 0.97321 -0.84989 0.72248 -0.59161 0.45790 -0.32199 0.18454 -0.04620 -0.09237 0.23049 -0.36750 0.50275 -0.63558 0.76537 -0.89148 1.01331 -1.13027 1.24181 -1.34739 1.44650 -1.53867 1.62345 -1.70043 1.76926 -1.82959 1.88113 -1.92365 1.95693 -1.98082 1.99520 -2.00000 0.00000 0.00000 ... ... ..... ....... ..TT..... ........... ......TT..... E134 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... ............. E135 2.00000 -1.99947 1.99787 -1.99520 1.99147 -1.98667 1.98082 -1.97391 1.96595 -1.95693 1.94688 -1.93578 1.92365 -1.91050 1.89632 -1.88113 1.86494 -1.84776 1.82959 -1.81044 1.79033 -1.76926 1.74724 -1.72430 1.70043 -1.67566 1.64999 -1.62345 1.59603 -1.56777 1.53867 -1.50875 1.47802 -1.44650 1.41421 -1.38117 1.34739 -1.31289 1.27769 -1.24181 1.20527 -1.16808 1.13027 -1.09186 1.05286 -1.01331 0.97321 -0.93259 0.89148 -0.84989 0.80784 -0.76537 0.72248 -0.67921 0.63558 -0.59161 0.54733 -0.50275 0.45790 -0.41281 0.36750 -0.32199 0.27631 -0.23049 0.18454 -0.13849 0.09237 -0.04620 0.00000 0.04620 -0.09237 0.13849 -0.18454 0.23049 -0.27631 0.32199 -0.36750 0.41281 -0.45790 0.50275 -0.54733 0.59161 -0.63558 0.67921 -0.72248 0.76537 -0.80784 0.84989 -0.89148 0.93259 -0.97321 1.01331 -1.05286 1.09186 -1.13027 1.16808 -1.20527 1.24181 -1.27769 1.31289 -1.34739 1.38117 -1.41421 1.44650 -1.47802 1.50875 -1.53867 1.56777 -1.59603 1.62345 -1.64999 1.67566 -1.70043 1.72430 -1.74724 1.76926 -1.79033 1.81044 -1.82959 1.84776 -1.86494 1.88113 -1.89632 1.91050 -1.92365 1.93578 -1.94688 1.95693 -1.96595 1.97391 -1.98082 1.98667 -1.99147 1.99520 -1.99787 1.99947 -2.00000 0.00000 0.00000 TT. ... ..TT. ....... ......TT. ........... ..........TT. Irrational character values: 1.999466416223 = 2*cos(2*π/272) = 2*cos(π/136) 1.997865949605 = 2*cos(4*π/272) = 2*cos(π/68) = √8+√2*√17−√17+√34−2*√17+2*√17+3*√17+√34−2*√17+2*√34+2*√17/2 1.995199454127 = 2*cos(6*π/272) = 2*cos(3*π/136) 1.991468352590 = 2*cos(8*π/272) = 2*cos(π/34) = √2*√17−√17+√34−2*√17+2*√17+3*√17+√34−2*√17+2*√34+2*√17/4 1.986674635848 = 2*cos(10*π/272) = 2*cos(5*π/136) 1.980820861750 = 2*cos(12*π/272) = 2*cos(3*π/68) = √8+√2*√17+√17+√34+2*√17+2*√17−3*√17+√34+2*√17−2*√34−2*√17/2 1.973910153776 = 2*cos(14*π/272) = 2*cos(7*π/136) 1.965946199368 = 2*cos(16*π/272) = 2*cos(π/17) = (1−√17+√34−2*√17+2*√17+3*√17+√34−2*√17+2*√34+2*√17)/8 1.956933247961 = 2*cos(18*π/272) = 2*cos(9*π/136) 1.946876108721 = 2*cos(20*π/272) = 2*cos(5*π/68) = √8+√2*√17+√17+√34+2*√17−2*√17−3*√17+√34+2*√17−2*√34−2*√17/2 1.935780147975 = 2*cos(22*π/272) = 2*cos(11*π/136) 1.923651286346 = 2*cos(24*π/272) = 2*cos(3*π/34) = √2*√17+√17+√34+2*√17+2*√17−3*√17+√34+2*√17−2*√34−2*√17/4 1.910495995598 = 2*cos(26*π/272) = 2*cos(13*π/136) 1.896321295182 = 2*cos(28*π/272) = 2*cos(7*π/68) = √8+√2*√17+√17−√34+2*√17+2*√17−3*√17−√34+2*√17+2*√34−2*√17/2 1.881134748487 = 2*cos(30*π/272) = 2*cos(15*π/136) 1.864944458809 = 2*cos(32*π/272) = 2*cos(2*π/17) = (−1+√17+√34−2*√17+2*√17+3*√17−√34−2*√17−2*√34+2*√17)/8 1.847759065023 = 2*cos(34*π/272) = 2*cos(π/8) = √2+√2 1.829587736976 = 2*cos(36*π/272) = 2*cos(9*π/68) = √8+√2*√17−√17−√34−2*√17+2*√17+3*√17−√34−2*√17−2*√34+2*√17/2 1.810440170595 = 2*cos(38*π/272) = 2*cos(19*π/136) 1.790326582710 = 2*cos(40*π/272) = 2*cos(5*π/34) = √2*√17+√17+√34+2*√17−2*√17−3*√17+√34+2*√17−2*√34−2*√17/4 1.769257705606 = 2*cos(42*π/272) = 2*cos(21*π/136) 1.747244781293 = 2*cos(44*π/272) = 2*cos(11*π/68) = √8+√2*√17+√17−√34+2*√17−2*√17−3*√17−√34+2*√17+2*√34−2*√17/2 1.724299555511 = 2*cos(46*π/272) = 2*cos(23*π/136) 1.700434271459 = 2*cos(48*π/272) = 2*cos(3*π/17) = (1+√17+√34+2*√17+2*√17−3*√17+√34+2*√17−2*√34−2*√17)/8 1.675661663267 = 2*cos(50*π/272) = 2*cos(25*π/136) 1.649994949197 = 2*cos(52*π/272) = 2*cos(13*π/68) = √8+√2*√17−√17+√34−2*√17−2*√17+3*√17+√34−2*√17+2*√34+2*√17/2 1.623447824589 = 2*cos(54*π/272) = 2*cos(27*π/136) 1.596034454560 = 2*cos(56*π/272) = 2*cos(7*π/34) = √2*√17+√17−√34+2*√17+2*√17−3*√17−√34+2*√17+2*√34−2*√17/4 1.567769466440 = 2*cos(58*π/272) = 2*cos(29*π/136) 1.538667941966 = 2*cos(60*π/272) = 2*cos(15*π/68) = √8+√2*√17−√17−√34−2*√17−2*√17+3*√17−√34−2*√17−2*√34+2*√17/2 1.508745409240 = 2*cos(62*π/272) = 2*cos(31*π/136) 1.478017834441 = 2*cos(64*π/272) = 2*cos(4*π/17) = (−1+√17−√34−2*√17+2*√17+3*√17+√34−2*√17+2*√34+2*√17)/8 1.446501613304 = 2*cos(66*π/272) = 2*cos(33*π/136) 1.414213562373 = 2*cos(68*π/272) = 2*cos(π/4) = √2 1.381170910028 = 2*cos(70*π/272) = 2*cos(35*π/136) 1.347391287293 = 2*cos(72*π/272) = 2*cos(9*π/34) = √2*√17−√17−√34−2*√17+2*√17+3*√17−√34−2*√17−2*√34+2*√17/4 1.312892718426 = 2*cos(74*π/272) = 2*cos(37*π/136) 1.277693611304 = 2*cos(76*π/272) = 2*cos(19*π/68) = √8−√2*√17−√17−√34−2*√17−2*√17+3*√17−√34−2*√17−2*√34+2*√17/2 1.241812747599 = 2*cos(78*π/272) = 2*cos(39*π/136) 1.205269272759 = 2*cos(80*π/272) = 2*cos(5*π/17) = (1+√17+√34+2*√17−2*√17−3*√17+√34+2*√17−2*√34−2*√17)/8 1.168082685787 = 2*cos(82*π/272) = 2*cos(41*π/136) 1.130272828845 = 2*cos(84*π/272) = 2*cos(21*π/68) = √8−√2*√17−√17+√34−2*√17−2*√17+3*√17+√34−2*√17+2*√34+2*√17/2 1.091859876658 = 2*cos(86*π/272) = 2*cos(43*π/136) 1.052864325755 = 2*cos(88*π/272) = 2*cos(11*π/34) = √2*√17+√17−√34+2*√17−2*√17−3*√17−√34+2*√17+2*√34−2*√17/4 1.013306983528 = 2*cos(90*π/272) = 2*cos(45*π/136) 0.973208957134 = 2*cos(92*π/272) = 2*cos(23*π/68) = √8−√2*√17+√17−√34+2*√17−2*√17−3*√17−√34+2*√17+2*√34−2*√17/2 0.932591642229 = 2*cos(94*π/272) = 2*cos(47*π/136) 0.891476711553 = 2*cos(96*π/272) = 2*cos(6*π/17) = (−1−√17+√34+2*√17+2*√17−3*√17−√34+2*√17+2*√34−2*√17)/8 0.849886103367 = 2*cos(98*π/272) = 2*cos(49*π/136) 0.807842009744 = 2*cos(100*π/272) = 2*cos(25*π/68) = √8−√2*√17−√17−√34−2*√17+2*√17+3*√17−√34−2*√17−2*√34+2*√17/2 0.765366864730 = 2*cos(102*π/272) = 2*cos(3*π/8) = √2−√2 0.722483332374 = 2*cos(104*π/272) = 2*cos(13*π/34) = √2*√17−√17+√34−2*√17−2*√17+3*√17+√34−2*√17+2*√34+2*√17/4 0.679214294633 = 2*cos(106*π/272) = 2*cos(53*π/136) 0.635582839164 = 2*cos(108*π/272) = 2*cos(27*π/68) = √8−√2*√17+√17−√34+2*√17+2*√17−3*√17−√34+2*√17+2*√34−2*√17/2 0.591612247003 = 2*cos(110*π/272) = 2*cos(55*π/136) 0.547325980144 = 2*cos(112*π/272) = 2*cos(7*π/17) = (1+√17−√34+2*√17+2*√17−3*√17−√34+2*√17+2*√34−2*√17)/8 0.502747669022 = 2*cos(114*π/272) = 2*cos(57*π/136) 0.457901099900 = 2*cos(116*π/272) = 2*cos(29*π/68) = √8−√2*√17+√17+√34+2*√17−2*√17−3*√17+√34+2*√17−2*√34−2*√17/2 0.412810202180 = 2*cos(118*π/272) = 2*cos(59*π/136) 0.367499035633 = 2*cos(120*π/272) = 2*cos(15*π/34) = √2*√17−√17−√34−2*√17−2*√17+3*√17−√34−2*√17−2*√34+2*√17/4 0.321991777563 = 2*cos(122*π/272) = 2*cos(61*π/136) 0.276312709904 = 2*cos(124*π/272) = 2*cos(31*π/68) = √8−√2*√17+√17+√34+2*√17+2*√17−3*√17+√34+2*√17−2*√34−2*√17/2 0.230486206265 = 2*cos(126*π/272) = 2*cos(63*π/136) 0.184536718927 = 2*cos(128*π/272) = 2*cos(8*π/17) = (−1+√17+√34−2*√17−2*√17+3*√17−√34−2*√17−2*√34+2*√17)/8 0.138488765788 = 2*cos(130*π/272) = 2*cos(65*π/136) 0.092366917291 = 2*cos(132*π/272) = 2*cos(33*π/68) = √8−√2*√17−√17+√34−2*√17+2*√17+3*√17+√34−2*√17+2*√34+2*√17/2 0.046195783306 = 2*cos(134*π/272) = 2*cos(67*π/136) Symmetry of Rotations and Cartesian products A1 d+g+i+k+m z2, z4, z6 A2 R Rz 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 E127 m E128 l E129 k+m E130 j+l E131 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)} E132 h+j+l {z((x2−y2)2−4x2y2), xyz(x2−y2)} E133 g+i+k+m {xz(x2−3y2), yz(3x2−y2)}, {xz3(x2−3y2), yz3(3x2−y2)} E134 f+h+j+l {z(x2−y2), xyz}, {z3(x2−y2), xyz3} E135 R+d+g+i+k+m {Rx, Ry}, {xz, yz}, {xz3, yz3}, {xz5, yz5} Notes: α The order of the D136d point group is 544, and the order of the principal axis (S272) is 272. The group has 139 irreducible representations. β The D136d point group is isomorphic to C272v and D272. γ The D136d point group is generated by two symmetry elements, S272 and either a perpendicular C2′ or a vertical σd. Also, the group may be generated from a C2′ plus a σd (some pairs will yield smaller groups, though; choosing a minimum angle is safe). 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 σd set of symmetry planes contains neither the xz nor the yz planes; but it contains the median plane (x+y)z. ζ The lowest nonvanishing multipole moment in D136d 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. Some of these algebraic expressions, however, are too complicated to be included in the list above. 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. Regular polygons of order 34,68,136 etc are easily derived from the regular 17-gon by successive halving of angles.
| D127d | ||
| D128d | ||
| C136 C136v C136h D136 D136h | D136d | S136 |
| D144d | ||
| D153d |
This Character Table for the D136d point group was created by Gernot Katzer.
For other groups and some explanations, see the Main Page.