Character Table for Point Group D340d

Character table for the D_340d point group

D340d E 2 S680 2 C340 2 S680^3 2 C170 2 S136 2 C340^3 2 S680^7 2 C85 2 S680^9 2 C68 2 S680^11 2 C170^3 2 S680^13 2 C340^7 2 S136^3 2 C85^2 2 S40 2 C340^9 2 S680^19 2 C34 2 S680^21 2 C340^11 2 S680^23 2 C85^3 2 S136^5 2 C340^13 2 S680^27 2 C170^7 2 S680^29 2 C68^3 2 S680^31 2 C85^4 2 S680^33 2 C20 2 S136^7 2 C170^9 2 S680^37 2 C340^19 2 S680^39 2 C17 2 S680^41 2 C340^21 2 S680^43 2 C170^11 2 S136^9 2 C340^23 2 S680^47 2 C85^6 2 S680^49 2 C68^5 2 S40^3 2 C170^13 2 S680^53 2 C340^27 2 S136^11 2 C85^7 2 S680^57 2 C340^29 2 S680^59 2 C34^3 2 S680^61 2 C340^31 2 S680^63 2 C85^8 2 S136^13 2 C340^33 2 S680^67 2 C10 2 S680^69 2 C68^7 2 S680^71 2 C85^9 2 S680^73 2 C340^37 2 S136^15 2 C170^19 2 S680^77 2 C340^39 2 S680^79 2 C17^2 2 S680^81 2 C340^41 2 S680^83 2 C170^21 2 S8 2 C340^43 2 S680^87 2 C85^11 2 S680^89 2 C68^9 2 S680^91 2 C170^23 2 S680^93 2 C340^47 2 S136^19 2 C85^12 2 S680^97 2 C340^49 2 S680^99 2 C34^5 2 S680^101 2 C20^3 2 S680^103 2 C85^13 2 S136^21 2 C340^53 2 S680^107 2 C170^27 2 S680^109 2 C68^11 2 S680^111 2 C85^14 2 S680^113 2 C340^57 2 S136^23 2 C170^29 2 S680^117 2 C340^59 2 S40^7 2 C17^3 2 S680^121 2 C340^61 2 S680^123 2 C170^31 2 S136^25 2 C340^63 2 S680^127 2 C85^16 2 S680^129 2 C68^13 2 S680^131 2 C170^33 2 S680^133 2 C340^67 2 S136^27 2 C5 2 S680^137 2 C340^69 2 S680^139 2 C34^7 2 S680^141 2 C340^71 2 S680^143 2 C85^18 2 S136^29 2 C340^73 2 S680^147 2 C170^37 2 S680^149 2 C68^15 2 S680^151 2 C85^19 2 S40^9 2 C340^77 2 S136^31 2 C170^39 2 S680^157 2 C340^79 2 S680^159 2 C17^4 2 S680^161 2 C340^81 2 S680^163 2 C170^41 2 S136^33 2 C340^83 2 S680^167 2 C85^21 2 S680^169 2 C4 2 S680^171 2 C170^43 2 S680^173 2 C340^87 2 S136^35 2 C85^22 2 S680^177 2 C340^89 2 S680^179 2 C34^9 2 S680^181 2 C340^91 2 S680^183 2 C85^23 2 S136^37 2 C340^93 2 S40^11 2 C170^47 2 S680^189 2 C68^19 2 S680^191 2 C85^24 2 S680^193 2 C340^97 2 S136^39 2 C170^49 2 S680^197 2 C340^99 2 S680^199 2 C17^5 2 S680^201 2 C340^101 2 S680^203 2 C10^3 2 S136^41 2 C340^103 2 S680^207 2 C85^26 2 S680^209 2 C68^21 2 S680^211 2 C170^53 2 S680^213 2 C340^107 2 S136^43 2 C85^27 2 S680^217 2 C340^109 2 S680^219 2 C34^11 2 S40^13 2 C340^111 2 S680^223 2 C85^28 2 S136^45 2 C340^113 2 S680^227 2 C170^57 2 S680^229 2 C68^23 2 S680^231 2 C85^29 2 S680^233 2 C340^117 2 S136^47 2 C170^59 2 S680^237 2 C20^7 2 S680^239 2 C17^6 2 S680^241 2 C340^121 2 S680^243 2 C170^61 2 S136^49 2 C340^123 2 S680^247 2 C85^31 2 S680^249 2 C68^25 2 S680^251 2 C170^63 2 S680^253 2 C340^127 2 S8^3 2 C85^32 2 S680^257 2 C340^129 2 S680^259 2 C34^13 2 S680^261 2 C340^131 2 S680^263 2 C85^33 2 S136^53 2 C340^133 2 S680^267 2 C170^67 2 S680^269 2 C68^27 2 S680^271 2 C5^2 2 S680^273 2 C340^137 2 S136^55 2 C170^69 2 S680^277 2 C340^139 2 S680^279 2 C17^7 2 S680^281 2 C340^141 2 S680^283 2 C170^71 2 S136^57 2 C340^143 2 S680^287 2 C85^36 2 S40^17 2 C68^29 2 S680^291 2 C170^73 2 S680^293 2 C340^147 2 S136^59 2 C85^37 2 S680^297 2 C340^149 2 S680^299 2 C34^15 2 S680^301 2 C340^151 2 S680^303 2 C85^38 2 S136^61 2 C20^9 2 S680^307 2 C170^77 2 S680^309 2 C68^31 2 S680^311 2 C85^39 2 S680^313 2 C340^157 2 S136^63 2 C170^79 2 S680^317 2 C340^159 2 S680^319 2 C17^8 2 S680^321 2 C340^161 2 S40^19 2 C170^81 2 S136^65 2 C340^163 2 S680^327 2 C85^41 2 S680^329 2 C68^33 2 S680^331 2 C170^83 2 S680^333 2 C340^167 2 S136^67 2 C85^42 2 S680^337 2 C340^169 2 S680^339 C2 340 C2' 340 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 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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 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -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 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -1.00000 1.00000 -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. ......... ........TT. ............. 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TT... ....... ....TT... ........... ........TT... 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TT..... ......... ....TT..... ............. 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TT....... ........... ....TT....... 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TT......... ............. 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TT........... E7 2.00000 1.99582 1.98329 1.96247 1.93344 1.89632 1.85128 1.79849 1.73818 1.67060 1.59603 1.51479 1.42722 1.33368 1.23456 1.13027 1.02126 0.90798 0.79090 0.67052 0.54733 0.42185 0.29460 0.16613 0.03696 -0.09237 -0.22131 -0.34932 -0.47587 -0.60043 -0.72248 -0.84151 -0.95702 -1.06853 -1.17557 -1.27769 -1.37447 -1.46551 -1.55041 -1.62883 -1.70043 -1.76493 -1.82205 -1.87154 -1.91321 -1.94688 -1.97240 -1.98968 -1.99863 -1.99923 -1.99147 -1.97538 -1.95102 -1.91851 -1.87798 -1.82959 -1.77355 -1.71009 -1.63948 -1.56202 -1.47802 -1.38784 -1.29186 -1.19047 -1.08411 -0.97321 -0.85824 -0.73968 -0.61803 -0.49380 -0.36750 -0.23966 -0.11082 0.01848 0.14771 0.27631 0.40376 0.52953 0.65308 0.77390 0.89148 1.00533 1.11498 1.21996 1.31985 1.41421 1.50266 1.58483 1.66037 1.72896 1.79033 1.84420 1.89037 1.92863 1.95882 1.98082 1.99454 1.99991 1.99693 1.98559 1.96595 1.93808 1.90211 1.85819 1.80649 1.74724 1.68069 1.60710 1.52680 1.44011 1.34739 1.24904 1.14547 1.03711 0.92441 0.80784 0.68790 0.56508 0.43989 0.31287 0.18454 0.05543 -0.07390 -0.20293 -0.33111 -0.45790 -0.58278 -0.70522 -0.82471 -0.94076 -1.05286 -1.16057 -1.26342 -1.36099 -1.45287 -1.53867 -1.61803 -1.69063 -1.75616 -1.81435 -1.86494 -1.90774 -1.94256 -1.96926 -1.98772 -1.99787 -1.99966 -1.99309 -1.97818 -1.95501 -1.92365 -1.88425 -1.83697 -1.78201 -1.71960 -1.64999 -1.57349 -1.49040 -1.40109 -1.30591 -1.20527 -1.09959 -0.98931 -0.87490 -0.75682 -0.63558 -0.51169 -0.38565 -0.25800 -0.12927 0.00000 0.12927 0.25800 0.38565 0.51169 0.63558 0.75682 0.87490 0.98931 1.09959 1.20527 1.30591 1.40109 1.49040 1.57349 1.64999 1.71960 1.78201 1.83697 1.88425 1.92365 1.95501 1.97818 1.99309 1.99966 1.99787 1.98772 1.96926 1.94256 1.90774 1.86494 1.81435 1.75616 1.69063 1.61803 1.53867 1.45287 1.36099 1.26342 1.16057 1.05286 0.94076 0.82471 0.70522 0.58278 0.45790 0.33111 0.20293 0.07390 -0.05543 -0.18454 -0.31287 -0.43989 -0.56508 -0.68790 -0.80784 -0.92441 -1.03711 -1.14547 -1.24904 -1.34739 -1.44011 -1.52680 -1.60710 -1.68069 -1.74724 -1.80649 -1.85819 -1.90211 -1.93808 -1.96595 -1.98559 -1.99693 -1.99991 -1.99454 -1.98082 -1.95882 -1.92863 -1.89037 -1.84420 -1.79033 -1.72896 -1.66037 -1.58483 -1.50266 -1.41421 -1.31985 -1.21996 -1.11498 -1.00533 -0.89148 -0.77390 -0.65308 -0.52953 -0.40376 -0.27631 -0.14771 -0.01848 0.11082 0.23966 0.36750 0.49380 0.61803 0.73968 0.85824 0.97321 1.08411 1.19047 1.29186 1.38784 1.47802 1.56202 1.63948 1.71009 1.77355 1.82959 1.87798 1.91851 1.95102 1.97538 1.99147 1.99923 1.99863 1.98968 1.97240 1.94688 1.91321 1.87154 1.82205 1.76493 1.70043 1.62883 1.55041 1.46551 1.37447 1.27769 1.17557 1.06853 0.95702 0.84151 0.72248 0.60043 0.47587 0.34932 0.22131 0.09237 -0.03696 -0.16613 -0.29460 -0.42185 -0.54733 -0.67052 -0.79090 -0.90798 -1.02126 -1.13027 -1.23456 -1.33368 -1.42722 -1.51479 -1.59603 -1.67060 -1.73818 -1.79849 -1.85128 -1.89632 -1.93344 -1.96247 -1.98329 -1.99582 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. 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E339 2.00000 -1.99991 1.99966 -1.99923 1.99863 -1.99787 1.99693 -1.99582 1.99454 -1.99309 1.99147 -1.98968 1.98772 -1.98559 1.98329 -1.98082 1.97818 -1.97538 1.97240 -1.96926 1.96595 -1.96247 1.95882 -1.95501 1.95102 -1.94688 1.94256 -1.93808 1.93344 -1.92863 1.92365 -1.91851 1.91321 -1.90774 1.90211 -1.89632 1.89037 -1.88425 1.87798 -1.87154 1.86494 -1.85819 1.85128 -1.84420 1.83697 -1.82959 1.82205 -1.81435 1.80649 -1.79849 1.79033 -1.78201 1.77355 -1.76493 1.75616 -1.74724 1.73818 -1.72896 1.71960 -1.71009 1.70043 -1.69063 1.68069 -1.67060 1.66037 -1.64999 1.63948 -1.62883 1.61803 -1.60710 1.59603 -1.58483 1.57349 -1.56202 1.55041 -1.53867 1.52680 -1.51479 1.50266 -1.49040 1.47802 -1.46551 1.45287 -1.44011 1.42722 -1.41421 1.40109 -1.38784 1.37447 -1.36099 1.34739 -1.33368 1.31985 -1.30591 1.29186 -1.27769 1.26342 -1.24904 1.23456 -1.21996 1.20527 -1.19047 1.17557 -1.16057 1.14547 -1.13027 1.11498 -1.09959 1.08411 -1.06853 1.05286 -1.03711 1.02126 -1.00533 0.98931 -0.97321 0.95702 -0.94076 0.92441 -0.90798 0.89148 -0.87490 0.85824 -0.84151 0.82471 -0.80784 0.79090 -0.77390 0.75682 -0.73968 0.72248 -0.70522 0.68790 -0.67052 0.65308 -0.63558 0.61803 -0.60043 0.58278 -0.56508 0.54733 -0.52953 0.51169 -0.49380 0.47587 -0.45790 0.43989 -0.42185 0.40376 -0.38565 0.36750 -0.34932 0.33111 -0.31287 0.29460 -0.27631 0.25800 -0.23966 0.22131 -0.20293 0.18454 -0.16613 0.14771 -0.12927 0.11082 -0.09237 0.07390 -0.05543 0.03696 -0.01848 0.00000 0.01848 -0.03696 0.05543 -0.07390 0.09237 -0.11082 0.12927 -0.14771 0.16613 -0.18454 0.20293 -0.22131 0.23966 -0.25800 0.27631 -0.29460 0.31287 -0.33111 0.34932 -0.36750 0.38565 -0.40376 0.42185 -0.43989 0.45790 -0.47587 0.49380 -0.51169 0.52953 -0.54733 0.56508 -0.58278 0.60043 -0.61803 0.63558 -0.65308 0.67052 -0.68790 0.70522 -0.72248 0.73968 -0.75682 0.77390 -0.79090 0.80784 -0.82471 0.84151 -0.85824 0.87490 -0.89148 0.90798 -0.92441 0.94076 -0.95702 0.97321 -0.98931 1.00533 -1.02126 1.03711 -1.05286 1.06853 -1.08411 1.09959 -1.11498 1.13027 -1.14547 1.16057 -1.17557 1.19047 -1.20527 1.21996 -1.23456 1.24904 -1.26342 1.27769 -1.29186 1.30591 -1.31985 1.33368 -1.34739 1.36099 -1.37447 1.38784 -1.40109 1.41421 -1.42722 1.44011 -1.45287 1.46551 -1.47802 1.49040 -1.50266 1.51479 -1.52680 1.53867 -1.55041 1.56202 -1.57349 1.58483 -1.59603 1.60710 -1.61803 1.62883 -1.63948 1.64999 -1.66037 1.67060 -1.68069 1.69063 -1.70043 1.71009 -1.71960 1.72896 -1.73818 1.74724 -1.75616 1.76493 -1.77355 1.78201 -1.79033 1.79849 -1.80649 1.81435 -1.82205 1.82959 -1.83697 1.84420 -1.85128 1.85819 -1.86494 1.87154 -1.87798 1.88425 -1.89037 1.89632 -1.90211 1.90774 -1.91321 1.91851 -1.92365 1.92863 -1.93344 1.93808 -1.94256 1.94688 -1.95102 1.95501 -1.95882 1.96247 -1.96595 1.96926 -1.97240 1.97538 -1.97818 1.98082 -1.98329 1.98559 -1.98772 1.98968 -1.99147 1.99309 -1.99454 1.99582 -1.99693 1.99787 -1.99863 1.99923 -1.99966 1.99991 -2.00000 0.00000 0.00000 TT. ... ..TT. ....... ......TT. ........... ..........TT. Irrational character values: 1.999914623407 = 2*cos(2*π/680) = 2*cos(π/340) 1.999658500916 = 2*cos(4*π/680) = 2*cos(π/170) 1.999231654395 = 2*cos(6*π/680) = 2*cos(3*π/340) 1.998634120286 = 2*cos(8*π/680) = 2*cos(π/85) 1.997865949605 = 2*cos(10*π/680) = 2*cos(π/68) = √8+√2*√17−√17+√34−2*√17+2*√17+3*√17+√34−2*√17+2*√34+2*√17/2 1.996927207935 = 2*cos(12*π/680) = 2*cos(3*π/170) 1.995817975423 = 2*cos(14*π/680) = 2*cos(7*π/340) 1.994538346772 = 2*cos(16*π/680) = 2*cos(2*π/85) 1.993088431231 = 2*cos(18*π/680) = 2*cos(9*π/340) 1.991468352590 = 2*cos(20*π/680) = 2*cos(π/34) = √2*√17−√17+√34−2*√17+2*√17+3*√17+√34−2*√17+2*√34+2*√17/4 1.989678249166 = 2*cos(22*π/680) = 2*cos(11*π/340) 1.987718273791 = 2*cos(24*π/680) = 2*cos(3*π/85) 1.985588593801 = 2*cos(26*π/680) = 2*cos(13*π/340) 1.983289391021 = 2*cos(28*π/680) = 2*cos(7*π/170) 1.980820861750 = 2*cos(30*π/680) = 2*cos(3*π/68) = √8+√2*√17+√17+√34+2*√17+2*√17−3*√17+√34+2*√17−2*√34−2*√17/2 1.978183216742 = 2*cos(32*π/680) = 2*cos(4*π/85) 1.975376681190 = 2*cos(34*π/680) = 2*cos(π/20) = (√2+√10+2*(√5−√5))/4 1.972401494707 = 2*cos(36*π/680) = 2*cos(9*π/170) 1.969257911303 = 2*cos(38*π/680) = 2*cos(19*π/340) 1.965946199368 = 2*cos(40*π/680) = 2*cos(π/17) = (1−√17+√34−2*√17+2*√17+3*√17+√34−2*√17+2*√34+2*√17)/8 1.962466641643 = 2*cos(42*π/680) = 2*cos(21*π/340) 1.958819535203 = 2*cos(44*π/680) = 2*cos(11*π/170) 1.955005191423 = 2*cos(46*π/680) = 2*cos(23*π/340) 1.951023935961 = 2*cos(48*π/680) = 2*cos(6*π/85) 1.946876108721 = 2*cos(50*π/680) = 2*cos(5*π/68) = √8+√2*√17+√17+√34+2*√17−2*√17−3*√17+√34+2*√17−2*√34−2*√17/2 1.942562063832 = 2*cos(52*π/680) = 2*cos(13*π/170) 1.938082169612 = 2*cos(54*π/680) = 2*cos(27*π/340) 1.933436808538 = 2*cos(56*π/680) = 2*cos(7*π/85) 1.928626377217 = 2*cos(58*π/680) = 2*cos(29*π/340) 1.923651286346 = 2*cos(60*π/680) = 2*cos(3*π/34) = √2*√17+√17+√34+2*√17+2*√17−3*√17+√34+2*√17−2*√34−2*√17/4 1.918511960681 = 2*cos(62*π/680) = 2*cos(31*π/340) 1.913208839001 = 2*cos(64*π/680) = 2*cos(8*π/85) 1.907742374068 = 2*cos(66*π/680) = 2*cos(33*π/340) 1.902113032590 = 2*cos(68*π/680) = 2*cos(π/10) = (√10+2*√5)/2 1.896321295182 = 2*cos(70*π/680) = 2*cos(7*π/68) = √8+√2*√17+√17−√34+2*√17+2*√17−3*√17−√34+2*√17+2*√34−2*√17/2 1.890367656322 = 2*cos(72*π/680) = 2*cos(9*π/85) 1.884252624311 = 2*cos(74*π/680) = 2*cos(37*π/340) 1.877976721230 = 2*cos(76*π/680) = 2*cos(19*π/170) 1.871540482895 = 2*cos(78*π/680) = 2*cos(39*π/340) 1.864944458809 = 2*cos(80*π/680) = 2*cos(2*π/17) = (−1+√17+√34−2*√17+2*√17+3*√17−√34−2*√17−2*√34+2*√17)/8 1.858189212118 = 2*cos(82*π/680) = 2*cos(41*π/340) 1.851275319563 = 2*cos(84*π/680) = 2*cos(21*π/170) 1.844203371428 = 2*cos(86*π/680) = 2*cos(43*π/340) 1.836973971492 = 2*cos(88*π/680) = 2*cos(11*π/85) 1.829587736976 = 2*cos(90*π/680) = 2*cos(9*π/68) = √8+√2*√17−√17−√34−2*√17+2*√17+3*√17−√34−2*√17−2*√34+2*√17/2 1.822045298492 = 2*cos(92*π/680) = 2*cos(23*π/170) 1.814347299988 = 2*cos(94*π/680) = 2*cos(47*π/340) 1.806494398692 = 2*cos(96*π/680) = 2*cos(12*π/85) 1.798487265059 = 2*cos(98*π/680) = 2*cos(49*π/340) 1.790326582710 = 2*cos(100*π/680) = 2*cos(5*π/34) = √2*√17+√17+√34+2*√17−2*√17−3*√17+√34+2*√17−2*√34−2*√17/4 1.782013048377 = 2*cos(102*π/680) = 2*cos(3*π/20) = (√10−√2+2*√5+√5)/4 1.773547371840 = 2*cos(104*π/680) = 2*cos(13*π/85) 1.764930275871 = 2*cos(106*π/680) = 2*cos(53*π/340) 1.756162496167 = 2*cos(108*π/680) = 2*cos(27*π/170) 1.747244781293 = 2*cos(110*π/680) = 2*cos(11*π/68) = √8+√2*√17+√17−√34+2*√17−2*√17−3*√17−√34+2*√17+2*√34−2*√17/2 1.738177892611 = 2*cos(112*π/680) = 2*cos(14*π/85) 1.728962604222 = 2*cos(114*π/680) = 2*cos(57*π/340) 1.719599702897 = 2*cos(116*π/680) = 2*cos(29*π/170) 1.710089988007 = 2*cos(118*π/680) = 2*cos(59*π/340) 1.700434271459 = 2*cos(120*π/680) = 2*cos(3*π/17) = (1+√17+√34+2*√17+2*√17−3*√17+√34+2*√17−2*√34−2*√17)/8 1.690633377627 = 2*cos(122*π/680) = 2*cos(61*π/340) 1.680688143276 = 2*cos(124*π/680) = 2*cos(31*π/170) 1.670599417497 = 2*cos(126*π/680) = 2*cos(63*π/340) 1.660368061631 = 2*cos(128*π/680) = 2*cos(16*π/85) 1.649994949197 = 2*cos(130*π/680) = 2*cos(13*π/68) = √8+√2*√17−√17+√34−2*√17−2*√17+3*√17+√34−2*√17+2*√34+2*√17/2 1.639480965814 = 2*cos(132*π/680) = 2*cos(33*π/170) 1.628827009133 = 2*cos(134*π/680) = 2*cos(67*π/340) 1.618033988750 = 2*cos(136*π/680) = 2*cos(π/5) = (√5+1)/2 1.607102826137 = 2*cos(138*π/680) = 2*cos(69*π/340) 1.596034454560 = 2*cos(140*π/680) = 2*cos(7*π/34) = √2*√17+√17−√34+2*√17+2*√17−3*√17−√34+2*√17+2*√34−2*√17/4 1.584829818999 = 2*cos(142*π/680) = 2*cos(71*π/340) 1.573489876067 = 2*cos(144*π/680) = 2*cos(18*π/85) 1.562015593930 = 2*cos(146*π/680) = 2*cos(73*π/340) 1.550407952222 = 2*cos(148*π/680) = 2*cos(37*π/170) 1.538667941966 = 2*cos(150*π/680) = 2*cos(15*π/68) = √8+√2*√17−√17−√34−2*√17−2*√17+3*√17−√34−2*√17−2*√34+2*√17/2 1.526796565482 = 2*cos(152*π/680) = 2*cos(19*π/85) 1.514794836309 = 2*cos(154*π/680) = 2*cos(77*π/340) 1.502663779114 = 2*cos(156*π/680) = 2*cos(39*π/170) 1.490404429604 = 2*cos(158*π/680) = 2*cos(79*π/340) 1.478017834441 = 2*cos(160*π/680) = 2*cos(4*π/17) = (−1+√17−√34−2*√17+2*√17+3*√17+√34−2*√17+2*√34+2*√17)/8 1.465505051151 = 2*cos(162*π/680) = 2*cos(81*π/340) 1.452867148032 = 2*cos(164*π/680) = 2*cos(41*π/170) 1.440105204066 = 2*cos(166*π/680) = 2*cos(83*π/340) 1.427220308824 = 2*cos(168*π/680) = 2*cos(21*π/85) 1.414213562373 = 2*cos(170*π/680) = 2*cos(π/4) = √2 1.401086075187 = 2*cos(172*π/680) = 2*cos(43*π/170) 1.387838968044 = 2*cos(174*π/680) = 2*cos(87*π/340) 1.374473371939 = 2*cos(176*π/680) = 2*cos(22*π/85) 1.360990427979 = 2*cos(178*π/680) = 2*cos(89*π/340) 1.347391287293 = 2*cos(180*π/680) = 2*cos(9*π/34) = √2*√17−√17−√34−2*√17+2*√17+3*√17−√34−2*√17−2*√34+2*√17/4 1.333677110929 = 2*cos(182*π/680) = 2*cos(91*π/340) 1.319849069757 = 2*cos(184*π/680) = 2*cos(23*π/85) 1.305908344368 = 2*cos(186*π/680) = 2*cos(93*π/340) 1.291856124974 = 2*cos(188*π/680) = 2*cos(47*π/170) 1.277693611304 = 2*cos(190*π/680) = 2*cos(19*π/68) = √8−√2*√17−√17−√34−2*√17−2*√17+3*√17−√34−2*√17−2*√34+2*√17/2 1.263422012507 = 2*cos(192*π/680) = 2*cos(24*π/85) 1.249042547042 = 2*cos(194*π/680) = 2*cos(97*π/340) 1.234556442580 = 2*cos(196*π/680) = 2*cos(49*π/170) 1.219964935894 = 2*cos(198*π/680) = 2*cos(99*π/340) 1.205269272759 = 2*cos(200*π/680) = 2*cos(5*π/17) = (1+√17+√34+2*√17−2*√17−3*√17+√34+2*√17−2*√34−2*√17)/8 1.190470707838 = 2*cos(202*π/680) = 2*cos(101*π/340) 1.175570504585 = 2*cos(204*π/680) = 2*cos(3*π/10) = (√10−2*√5)/2 1.160569935127 = 2*cos(206*π/680) = 2*cos(103*π/340) 1.145470280161 = 2*cos(208*π/680) = 2*cos(26*π/85) 1.130272828845 = 2*cos(210*π/680) = 2*cos(21*π/68) = √8−√2*√17−√17+√34−2*√17−2*√17+3*√17+√34−2*√17+2*√34+2*√17/2 1.114978878686 = 2*cos(212*π/680) = 2*cos(53*π/170) 1.099589735428 = 2*cos(214*π/680) = 2*cos(107*π/340) 1.084106712945 = 2*cos(216*π/680) = 2*cos(27*π/85) 1.068531133124 = 2*cos(218*π/680) = 2*cos(109*π/340) 1.052864325755 = 2*cos(220*π/680) = 2*cos(11*π/34) = √2*√17+√17−√34+2*√17−2*√17−3*√17−√34+2*√17+2*√34−2*√17/4 1.037107628416 = 2*cos(222*π/680) = 2*cos(111*π/340) 1.021262386362 = 2*cos(224*π/680) = 2*cos(28*π/85) 1.005329952404 = 2*cos(226*π/680) = 2*cos(113*π/340) 0.989311686800 = 2*cos(228*π/680) = 2*cos(57*π/170) 0.973208957134 = 2*cos(230*π/680) = 2*cos(23*π/68) = √8−√2*√17+√17−√34+2*√17−2*√17−3*√17−√34+2*√17+2*√34−2*√17/2 0.957023138203 = 2*cos(232*π/680) = 2*cos(29*π/85) 0.940755611896 = 2*cos(234*π/680) = 2*cos(117*π/340) 0.924407767081 = 2*cos(236*π/680) = 2*cos(59*π/170) 0.907980999479 = 2*cos(238*π/680) = 2*cos(7*π/20) = (√2−√10+2*√5+√5)/4 0.891476711553 = 2*cos(240*π/680) = 2*cos(6*π/17) = (−1−√17+√34+2*√17+2*√17−3*√17−√34+2*√17+2*√34−2*√17)/8 0.874896312382 = 2*cos(242*π/680) = 2*cos(121*π/340) 0.858241217545 = 2*cos(244*π/680) = 2*cos(61*π/170) 0.841512848997 = 2*cos(246*π/680) = 2*cos(123*π/340) 0.824712634948 = 2*cos(248*π/680) = 2*cos(31*π/85) 0.807842009744 = 2*cos(250*π/680) = 2*cos(25*π/68) = √8−√2*√17−√17−√34−2*√17+2*√17+3*√17−√34−2*√17−2*√34+2*√17/2 0.790902413741 = 2*cos(252*π/680) = 2*cos(63*π/170) 0.773895293185 = 2*cos(254*π/680) = 2*cos(127*π/340) 0.756822100085 = 2*cos(256*π/680) = 2*cos(32*π/85) 0.739684292092 = 2*cos(258*π/680) = 2*cos(129*π/340) 0.722483332374 = 2*cos(260*π/680) = 2*cos(13*π/34) = √2*√17−√17+√34−2*√17−2*√17+3*√17+√34−2*√17+2*√34+2*√17/4 0.705220689491 = 2*cos(262*π/680) = 2*cos(131*π/340) 0.687897837268 = 2*cos(264*π/680) = 2*cos(33*π/85) 0.670516254671 = 2*cos(266*π/680) = 2*cos(133*π/340) 0.653077425680 = 2*cos(268*π/680) = 2*cos(67*π/170) 0.635582839164 = 2*cos(270*π/680) = 2*cos(27*π/68) = √8−√2*√17+√17−√34+2*√17+2*√17−3*√17−√34+2*√17+2*√34−2*√17/2 0.618033988750 = 2*cos(272*π/680) = 2*cos(2*π/5) = (√5−1)/2 0.600432372700 = 2*cos(274*π/680) = 2*cos(137*π/340) 0.582779493779 = 2*cos(276*π/680) = 2*cos(69*π/170) 0.565076859130 = 2*cos(278*π/680) = 2*cos(139*π/340) 0.547325980144 = 2*cos(280*π/680) = 2*cos(7*π/17) = (1+√17−√34+2*√17+2*√17−3*√17−√34+2*√17+2*√34−2*√17)/8 0.529528372331 = 2*cos(282*π/680) = 2*cos(141*π/340) 0.511685555189 = 2*cos(284*π/680) = 2*cos(71*π/170) 0.493799052077 = 2*cos(286*π/680) = 2*cos(143*π/340) 0.475870390085 = 2*cos(288*π/680) = 2*cos(36*π/85) 0.457901099900 = 2*cos(290*π/680) = 2*cos(29*π/68) = √8−√2*√17+√17+√34+2*√17−2*√17−3*√17+√34+2*√17−2*√34−2*√17/2 0.439892715679 = 2*cos(292*π/680) = 2*cos(73*π/170) 0.421846774917 = 2*cos(294*π/680) = 2*cos(147*π/340) 0.403764818314 = 2*cos(296*π/680) = 2*cos(37*π/85) 0.385648389646 = 2*cos(298*π/680) = 2*cos(149*π/340) 0.367499035633 = 2*cos(300*π/680) = 2*cos(15*π/34) = √2*√17−√17−√34−2*√17−2*√17+3*√17−√34−2*√17−2*√34+2*√17/4 0.349318305804 = 2*cos(302*π/680) = 2*cos(151*π/340) 0.331107752368 = 2*cos(304*π/680) = 2*cos(38*π/85) 0.312868930080 = 2*cos(306*π/680) = 2*cos(9*π/20) = (√2+√10−2*(√5−√5))/4 0.294603396109 = 2*cos(308*π/680) = 2*cos(77*π/170) 0.276312709904 = 2*cos(310*π/680) = 2*cos(31*π/68) = √8−√2*√17+√17+√34+2*√17+2*√17−3*√17+√34+2*√17−2*√34−2*√17/2 0.257998433060 = 2*cos(312*π/680) = 2*cos(39*π/85) 0.239662129190 = 2*cos(314*π/680) = 2*cos(157*π/340) 0.221305363783 = 2*cos(316*π/680) = 2*cos(79*π/170) 0.202929704078 = 2*cos(318*π/680) = 2*cos(159*π/340) 0.184536718927 = 2*cos(320*π/680) = 2*cos(8*π/17) = (−1+√17+√34−2*√17−2*√17+3*√17−√34−2*√17−2*√34+2*√17)/8 0.166127978659 = 2*cos(322*π/680) = 2*cos(161*π/340) 0.147705054950 = 2*cos(324*π/680) = 2*cos(81*π/170) 0.129269520686 = 2*cos(326*π/680) = 2*cos(163*π/340) 0.110822949832 = 2*cos(328*π/680) = 2*cos(41*π/85) 0.092366917291 = 2*cos(330*π/680) = 2*cos(33*π/68) = √8−√2*√17−√17+√34−2*√17+2*√17+3*√17+√34−2*√17+2*√34+2*√17/2 0.073902998778 = 2*cos(332*π/680) = 2*cos(83*π/170) 0.055432770679 = 2*cos(334*π/680) = 2*cos(167*π/340) 0.036957809918 = 2*cos(336*π/680) = 2*cos(42*π/85) 0.018479693826 = 2*cos(338*π/680) = 2*cos(169*π/340) 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 E331 m E332 l E333 k+m E334 j+l E335 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)} E336 h+j+l {z((x²−y²)²−4x²y²), xyz(x²−y²)} E337 g+i+k+m {xz(x²−3y²), yz(3x²−y²)}, {xz³(x²−3y²), yz³(3x²−y²)} E338 f+h+j+l {z(x²−y²), xyz}, {z³(x²−y²), xyz³} E339 R+d+g+i+k+m {R_x, R_y}, {xz, yz}, {xz³, yz³}, {xz⁵, yz⁵} Notes: α The order of the D_340d point group is 1360, and the order of the principal axis (S₆₈₀) is 680. The group has 343 irreducible representations. β The D_340d point group is isomorphic to C_680v and D₆₈₀. γ The D_340d point group is generated by two symmetry elements, S₆₈₀ 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 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; but it contains the median plane (x+y)z. ζ The lowest nonvanishing multipole moment in D_340d 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 The fact that the regular pentagon is constructible is known since antiquity; Eukleides already discovered a construction for it. The double cosine of 2π/5 is equal to the reciprocal of the Golden Ratio of (1+√5)/2 = 1.61803. λ 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 85-gon can be derived from the regular 17-gon by overlaying it with a regular pentagon. Further halving of angles yields regular polygons of order 170,340,640 etc.

This Character Table for the D_340d point group was created by Gernot Katzer.

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

	D_256d
	D_257d
C₃₄₀ C_340v C_340h D₃₄₀ D_340h	D_340d	S₃₄₀
	D_512d
	D_666d

Character table for the D340d point group

Character table for the D_340d point group