Character Table for Point Group C666v

Character table for the C_666v point group

C666v E 2 C666 2 C333 2 C222 2 C333^2 2 C666^5 2 C111 2 C666^7 2 C333^4 2 C74 2 C333^5 2 C666^11 2 C111^2 2 C666^13 2 C333^7 2 C222^5 2 C333^8 2 C666^17 2 C37 2 C666^19 2 C333^10 2 C222^7 2 C333^11 2 C666^23 2 C111^4 2 C666^25 2 C333^13 2 C74^3 2 C333^14 2 C666^29 2 C111^5 2 C666^31 2 C333^16 2 C222^11 2 C333^17 2 C666^35 2 C37^2 2 C18 2 C333^19 2 C222^13 2 C333^20 2 C666^41 2 C111^7 2 C666^43 2 C333^22 2 C74^5 2 C333^23 2 C666^47 2 C111^8 2 C666^49 2 C333^25 2 C222^17 2 C333^26 2 C666^53 2 C37^3 2 C666^55 2 C333^28 2 C222^19 2 C333^29 2 C666^59 2 C111^10 2 C666^61 2 C333^31 2 C74^7 2 C333^32 2 C666^65 2 C111^11 2 C666^67 2 C333^34 2 C222^23 2 C333^35 2 C666^71 2 C37^4 2 C666^73 2 C9 2 C222^25 2 C333^38 2 C666^77 2 C111^13 2 C666^79 2 C333^40 2 C74^9 2 C333^41 2 C666^83 2 C111^14 2 C666^85 2 C333^43 2 C222^29 2 C333^44 2 C666^89 2 C37^5 2 C666^91 2 C333^46 2 C222^31 2 C333^47 2 C666^95 2 C111^16 2 C666^97 2 C333^49 2 C74^11 2 C333^50 2 C666^101 2 C111^17 2 C666^103 2 C333^52 2 C222^35 2 C333^53 2 C666^107 2 C37^6 2 C666^109 2 C333^55 2 C6 2 C333^56 2 C666^113 2 C111^19 2 C666^115 2 C333^58 2 C74^13 2 C333^59 2 C666^119 2 C111^20 2 C666^121 2 C333^61 2 C222^41 2 C333^62 2 C666^125 2 C37^7 2 C666^127 2 C333^64 2 C222^43 2 C333^65 2 C666^131 2 C111^22 2 C666^133 2 C333^67 2 C74^15 2 C333^68 2 C666^137 2 C111^23 2 C666^139 2 C333^70 2 C222^47 2 C333^71 2 C666^143 2 C37^8 2 C666^145 2 C333^73 2 C222^49 2 C9^2 2 C666^149 2 C111^25 2 C666^151 2 C333^76 2 C74^17 2 C333^77 2 C666^155 2 C111^26 2 C666^157 2 C333^79 2 C222^53 2 C333^80 2 C666^161 2 C37^9 2 C666^163 2 C333^82 2 C222^55 2 C333^83 2 C666^167 2 C111^28 2 C666^169 2 C333^85 2 C74^19 2 C333^86 2 C666^173 2 C111^29 2 C666^175 2 C333^88 2 C222^59 2 C333^89 2 C666^179 2 C37^10 2 C666^181 2 C333^91 2 C222^61 2 C333^92 2 C18^5 2 C111^31 2 C666^187 2 C333^94 2 C74^21 2 C333^95 2 C666^191 2 C111^32 2 C666^193 2 C333^97 2 C222^65 2 C333^98 2 C666^197 2 C37^11 2 C666^199 2 C333^100 2 C222^67 2 C333^101 2 C666^203 2 C111^34 2 C666^205 2 C333^103 2 C74^23 2 C333^104 2 C666^209 2 C111^35 2 C666^211 2 C333^106 2 C222^71 2 C333^107 2 C666^215 2 C37^12 2 C666^217 2 C333^109 2 C222^73 2 C333^110 2 C666^221 2 C3 2 C666^223 2 C333^112 2 C74^25 2 C333^113 2 C666^227 2 C111^38 2 C666^229 2 C333^115 2 C222^77 2 C333^116 2 C666^233 2 C37^13 2 C666^235 2 C333^118 2 C222^79 2 C333^119 2 C666^239 2 C111^40 2 C666^241 2 C333^121 2 C74^27 2 C333^122 2 C666^245 2 C111^41 2 C666^247 2 C333^124 2 C222^83 2 C333^125 2 C666^251 2 C37^14 2 C666^253 2 C333^127 2 C222^85 2 C333^128 2 C666^257 2 C111^43 2 C18^7 2 C333^130 2 C74^29 2 C333^131 2 C666^263 2 C111^44 2 C666^265 2 C333^133 2 C222^89 2 C333^134 2 C666^269 2 C37^15 2 C666^271 2 C333^136 2 C222^91 2 C333^137 2 C666^275 2 C111^46 2 C666^277 2 C333^139 2 C74^31 2 C333^140 2 C666^281 2 C111^47 2 C666^283 2 C333^142 2 C222^95 2 C333^143 2 C666^287 2 C37^16 2 C666^289 2 C333^145 2 C222^97 2 C333^146 2 C666^293 2 C111^49 2 C666^295 2 C9^4 2 C74^33 2 C333^149 2 C666^299 2 C111^50 2 C666^301 2 C333^151 2 C222^101 2 C333^152 2 C666^305 2 C37^17 2 C666^307 2 C333^154 2 C222^103 2 C333^155 2 C666^311 2 C111^52 2 C666^313 2 C333^157 2 C74^35 2 C333^158 2 C666^317 2 C111^53 2 C666^319 2 C333^160 2 C222^107 2 C333^161 2 C666^323 2 C37^18 2 C666^325 2 C333^163 2 C222^109 2 C333^164 2 C666^329 2 C111^55 2 C666^331 2 C333^166 C2 333 sv 333 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 ... ... ..... ....... ......... ........... ............. 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 ... ... ..... ....... ......... ........... ............. 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TT. ..TT. ....TT. ......TT. ........TT. ..........TT. 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TT... ..TT... ....TT... ......TT... ........TT... 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TT..... ..TT..... ....TT..... ......TT..... 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TT....... ..TT....... ....TT....... 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TT......... ..TT......... 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E332 2.00000 -1.99991 1.99964 -1.99920 1.99858 -1.99778 1.99680 -1.99564 1.99431 -1.99279 1.99111 -1.98924 1.98720 -1.98498 1.98258 -1.98001 1.97726 -1.97433 1.97123 -1.96796 1.96450 -1.96088 1.95708 -1.95310 1.94895 -1.94463 1.94013 -1.93547 1.93063 -1.92561 1.92043 -1.91507 1.90955 -1.90385 1.89799 -1.89196 1.88575 -1.87939 1.87285 -1.86614 1.85928 -1.85224 1.84504 -1.83768 1.83015 -1.82246 1.81460 -1.80659 1.79841 -1.79008 1.78158 -1.77293 1.76412 -1.75515 1.74603 -1.73675 1.72731 -1.71773 1.70799 -1.69809 1.68805 -1.67785 1.66751 -1.65702 1.64638 -1.63559 1.62466 -1.61359 1.60237 -1.59100 1.57950 -1.56786 1.55607 -1.54415 1.53209 -1.51989 1.50756 -1.49509 1.48250 -1.46977 1.45690 -1.44391 1.43079 -1.41755 1.40417 -1.39067 1.37705 -1.36331 1.34944 -1.33545 1.32135 -1.30713 1.29279 -1.27833 1.26377 -1.24909 1.23429 -1.21939 1.20438 -1.18927 1.17404 -1.15872 1.14329 -1.12775 1.11212 -1.09639 1.08056 -1.06463 1.04861 -1.03250 1.01630 -1.00000 0.98362 -0.96714 0.95058 -0.93394 0.91722 -0.90041 0.88352 -0.86655 0.84951 -0.83239 0.81520 -0.79793 0.78059 -0.76319 0.74571 -0.72817 0.71057 -0.69290 0.67517 -0.65738 0.63953 -0.62162 0.60366 -0.58565 0.56758 -0.54946 0.53129 -0.51308 0.49482 -0.47652 0.45817 -0.43978 0.42136 -0.40290 0.38440 -0.36586 0.34730 -0.32870 0.31007 -0.29142 0.27274 -0.25404 0.23531 -0.21656 0.19779 -0.17901 0.16021 -0.14140 0.12257 -0.10373 0.08488 -0.06603 0.04717 -0.02830 0.00943 0.00943 -0.02830 0.04717 -0.06603 0.08488 -0.10373 0.12257 -0.14140 0.16021 -0.17901 0.19779 -0.21656 0.23531 -0.25404 0.27274 -0.29142 0.31007 -0.32870 0.34730 -0.36586 0.38440 -0.40290 0.42136 -0.43978 0.45817 -0.47652 0.49482 -0.51308 0.53129 -0.54946 0.56758 -0.58565 0.60366 -0.62162 0.63953 -0.65738 0.67517 -0.69290 0.71057 -0.72817 0.74571 -0.76319 0.78059 -0.79793 0.81520 -0.83239 0.84951 -0.86655 0.88352 -0.90041 0.91722 -0.93394 0.95058 -0.96714 0.98362 -1.00000 1.01630 -1.03250 1.04861 -1.06463 1.08056 -1.09639 1.11212 -1.12775 1.14329 -1.15872 1.17404 -1.18927 1.20438 -1.21939 1.23429 -1.24909 1.26377 -1.27833 1.29279 -1.30713 1.32135 -1.33545 1.34944 -1.36331 1.37705 -1.39067 1.40417 -1.41755 1.43079 -1.44391 1.45690 -1.46977 1.48250 -1.49509 1.50756 -1.51989 1.53209 -1.54415 1.55607 -1.56786 1.57950 -1.59100 1.60237 -1.61359 1.62466 -1.63559 1.64638 -1.65702 1.66751 -1.67785 1.68805 -1.69809 1.70799 -1.71773 1.72731 -1.73675 1.74603 -1.75515 1.76412 -1.77293 1.78158 -1.79008 1.79841 -1.80659 1.81460 -1.82246 1.83015 -1.83768 1.84504 -1.85224 1.85928 -1.86614 1.87285 -1.87939 1.88575 -1.89196 1.89799 -1.90385 1.90955 -1.91507 1.92043 -1.92561 1.93063 -1.93547 1.94013 -1.94463 1.94895 -1.95310 1.95708 -1.96088 1.96450 -1.96796 1.97123 -1.97433 1.97726 -1.98001 1.98258 -1.98498 1.98720 -1.98924 1.99111 -1.99279 1.99431 -1.99564 1.99680 -1.99778 1.99858 -1.99920 1.99964 -1.99991 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. Irrational character values: 1.999910996301 = 2*cos(2*π/666) = 2*cos(π/333) 1.999643993125 = 2*cos(4*π/666) = 2*cos(2*π/333) 1.999199014237 = 2*cos(6*π/666) = 2*cos(π/111) 1.998576099241 = 2*cos(8*π/666) = 2*cos(4*π/333) 1.997775303579 = 2*cos(10*π/666) = 2*cos(5*π/333) 1.996796698525 = 2*cos(12*π/666) = 2*cos(2*π/111) 1.995640371178 = 2*cos(14*π/666) = 2*cos(7*π/333) 1.994306424456 = 2*cos(16*π/666) = 2*cos(8*π/333) 1.992794977085 = 2*cos(18*π/666) = 2*cos(π/37) 1.991106163589 = 2*cos(20*π/666) = 2*cos(10*π/333) 1.989240134280 = 2*cos(22*π/666) = 2*cos(11*π/333) 1.987197055239 = 2*cos(24*π/666) = 2*cos(4*π/111) 1.984977108310 = 2*cos(26*π/666) = 2*cos(13*π/333) 1.982580491076 = 2*cos(28*π/666) = 2*cos(14*π/333) 1.980007416844 = 2*cos(30*π/666) = 2*cos(5*π/111) 1.977258114627 = 2*cos(32*π/666) = 2*cos(16*π/333) 1.974332829124 = 2*cos(34*π/666) = 2*cos(17*π/333) 1.971231820695 = 2*cos(36*π/666) = 2*cos(2*π/37) 1.967955365343 = 2*cos(38*π/666) = 2*cos(19*π/333) 1.964503754683 = 2*cos(40*π/666) = 2*cos(20*π/333) 1.960877295923 = 2*cos(42*π/666) = 2*cos(7*π/111) 1.957076311829 = 2*cos(44*π/666) = 2*cos(22*π/333) 1.953101140704 = 2*cos(46*π/666) = 2*cos(23*π/333) 1.948952136352 = 2*cos(48*π/666) = 2*cos(8*π/111) 1.944629668051 = 2*cos(50*π/666) = 2*cos(25*π/333) 1.940134120516 = 2*cos(52*π/666) = 2*cos(26*π/333) 1.935465893867 = 2*cos(54*π/666) = 2*cos(3*π/37) 1.930625403594 = 2*cos(56*π/666) = 2*cos(28*π/333) 1.925613080518 = 2*cos(58*π/666) = 2*cos(29*π/333) 1.920429370755 = 2*cos(60*π/666) = 2*cos(10*π/111) 1.915074735674 = 2*cos(62*π/666) = 2*cos(31*π/333) 1.909549651858 = 2*cos(64*π/666) = 2*cos(32*π/333) 1.903854611058 = 2*cos(66*π/666) = 2*cos(11*π/111) 1.897990120156 = 2*cos(68*π/666) = 2*cos(34*π/333) 1.891956701111 = 2*cos(70*π/666) = 2*cos(35*π/333) 1.885754890922 = 2*cos(72*π/666) = 2*cos(4*π/37) 1.879385241572 = 2*cos(74*π/666) = 2*cos(π/9) 1.872848319983 = 2*cos(76*π/666) = 2*cos(38*π/333) 1.866144707965 = 2*cos(78*π/666) = 2*cos(13*π/111) 1.859275002166 = 2*cos(80*π/666) = 2*cos(40*π/333) 1.852239814013 = 2*cos(82*π/666) = 2*cos(41*π/333) 1.845039769665 = 2*cos(84*π/666) = 2*cos(14*π/111) 1.837675509952 = 2*cos(86*π/666) = 2*cos(43*π/333) 1.830147690322 = 2*cos(88*π/666) = 2*cos(44*π/333) 1.822456980776 = 2*cos(90*π/666) = 2*cos(5*π/37) 1.814604065818 = 2*cos(92*π/666) = 2*cos(46*π/333) 1.806589644386 = 2*cos(94*π/666) = 2*cos(47*π/333) 1.798414429792 = 2*cos(96*π/666) = 2*cos(16*π/111) 1.790079149661 = 2*cos(98*π/666) = 2*cos(49*π/333) 1.781584545864 = 2*cos(100*π/666) = 2*cos(50*π/333) 1.772931374452 = 2*cos(102*π/666) = 2*cos(17*π/111) 1.764120405590 = 2*cos(104*π/666) = 2*cos(52*π/333) 1.755152423485 = 2*cos(106*π/666) = 2*cos(53*π/333) 1.746028226322 = 2*cos(108*π/666) = 2*cos(6*π/37) 1.736748626189 = 2*cos(110*π/666) = 2*cos(55*π/333) 1.727314449003 = 2*cos(112*π/666) = 2*cos(56*π/333) 1.717726534441 = 2*cos(114*π/666) = 2*cos(19*π/111) 1.707985735863 = 2*cos(116*π/666) = 2*cos(58*π/333) 1.698092920237 = 2*cos(118*π/666) = 2*cos(59*π/333) 1.688048968060 = 2*cos(120*π/666) = 2*cos(20*π/111) 1.677854773280 = 2*cos(122*π/666) = 2*cos(61*π/333) 1.667511243218 = 2*cos(124*π/666) = 2*cos(62*π/333) 1.657019298488 = 2*cos(126*π/666) = 2*cos(7*π/37) 1.646379872910 = 2*cos(128*π/666) = 2*cos(64*π/333) 1.635593913433 = 2*cos(130*π/666) = 2*cos(65*π/333) 1.624662380048 = 2*cos(132*π/666) = 2*cos(22*π/111) 1.613586245701 = 2*cos(134*π/666) = 2*cos(67*π/333) 1.602366496209 = 2*cos(136*π/666) = 2*cos(68*π/333) 1.591004130171 = 2*cos(138*π/666) = 2*cos(23*π/111) 1.579500158881 = 2*cos(140*π/666) = 2*cos(70*π/333) 1.567855606233 = 2*cos(142*π/666) = 2*cos(71*π/333) 1.556071508637 = 2*cos(144*π/666) = 2*cos(8*π/37) 1.544148914920 = 2*cos(146*π/666) = 2*cos(73*π/333) 1.532088886238 = 2*cos(148*π/666) = 2*cos(2*π/9) 1.519892495977 = 2*cos(150*π/666) = 2*cos(25*π/111) 1.507560829662 = 2*cos(152*π/666) = 2*cos(76*π/333) 1.495094984857 = 2*cos(154*π/666) = 2*cos(77*π/333) 1.482496071067 = 2*cos(156*π/666) = 2*cos(26*π/111) 1.469765209643 = 2*cos(158*π/666) = 2*cos(79*π/333) 1.456903533678 = 2*cos(160*π/666) = 2*cos(80*π/333) 1.443912187909 = 2*cos(162*π/666) = 2*cos(9*π/37) 1.430792328614 = 2*cos(164*π/666) = 2*cos(82*π/333) 1.417545123510 = 2*cos(166*π/666) = 2*cos(83*π/333) 1.404171751645 = 2*cos(168*π/666) = 2*cos(28*π/111) 1.390673403301 = 2*cos(170*π/666) = 2*cos(85*π/333) 1.377051279879 = 2*cos(172*π/666) = 2*cos(86*π/333) 1.363306593799 = 2*cos(174*π/666) = 2*cos(29*π/111) 1.349440568389 = 2*cos(176*π/666) = 2*cos(88*π/333) 1.335454437777 = 2*cos(178*π/666) = 2*cos(89*π/333) 1.321349446780 = 2*cos(180*π/666) = 2*cos(10*π/37) 1.307126850794 = 2*cos(182*π/666) = 2*cos(91*π/333) 1.292787915684 = 2*cos(184*π/666) = 2*cos(92*π/333) 1.278333917666 = 2*cos(186*π/666) = 2*cos(31*π/111) 1.263766143201 = 2*cos(188*π/666) = 2*cos(94*π/333) 1.249085888874 = 2*cos(190*π/666) = 2*cos(95*π/333) 1.234294461283 = 2*cos(192*π/666) = 2*cos(32*π/111) 1.219393176919 = 2*cos(194*π/666) = 2*cos(97*π/333) 1.204383362051 = 2*cos(196*π/666) = 2*cos(98*π/333) 1.189266352609 = 2*cos(198*π/666) = 2*cos(11*π/37) 1.174043494062 = 2*cos(200*π/666) = 2*cos(100*π/333) 1.158716141301 = 2*cos(202*π/666) = 2*cos(101*π/333) 1.143285658517 = 2*cos(204*π/666) = 2*cos(34*π/111) 1.127753419080 = 2*cos(206*π/666) = 2*cos(103*π/333) 1.112120805418 = 2*cos(208*π/666) = 2*cos(104*π/333) 1.096389208889 = 2*cos(210*π/666) = 2*cos(35*π/111) 1.080560029666 = 2*cos(212*π/666) = 2*cos(106*π/333) 1.064634676602 = 2*cos(214*π/666) = 2*cos(107*π/333) 1.048614567114 = 2*cos(216*π/666) = 2*cos(12*π/37) 1.032501127051 = 2*cos(218*π/666) = 2*cos(109*π/333) 1.016295790568 = 2*cos(220*π/666) = 2*cos(110*π/333) 0.983615205733 = 2*cos(224*π/666) = 2*cos(112*π/333) 0.967142866074 = 2*cos(226*π/666) = 2*cos(113*π/333) 0.950584447122 = 2*cos(228*π/666) = 2*cos(38*π/111) 0.933941422638 = 2*cos(230*π/666) = 2*cos(115*π/333) 0.917215273913 = 2*cos(232*π/666) = 2*cos(116*π/333) 0.900407489635 = 2*cos(234*π/666) = 2*cos(13*π/37) 0.883519565760 = 2*cos(236*π/666) = 2*cos(118*π/333) 0.866553005376 = 2*cos(238*π/666) = 2*cos(119*π/333) 0.849509318568 = 2*cos(240*π/666) = 2*cos(40*π/111) 0.832390022289 = 2*cos(242*π/666) = 2*cos(121*π/333) 0.815196640218 = 2*cos(244*π/666) = 2*cos(122*π/333) 0.797930702631 = 2*cos(246*π/666) = 2*cos(41*π/111) 0.780593746259 = 2*cos(248*π/666) = 2*cos(124*π/333) 0.763187314157 = 2*cos(250*π/666) = 2*cos(125*π/333) 0.745712955561 = 2*cos(252*π/666) = 2*cos(14*π/37) 0.728172225753 = 2*cos(254*π/666) = 2*cos(127*π/333) 0.710566685923 = 2*cos(256*π/666) = 2*cos(128*π/333) 0.692897903029 = 2*cos(258*π/666) = 2*cos(43*π/111) 0.675167449660 = 2*cos(260*π/666) = 2*cos(130*π/333) 0.657376903889 = 2*cos(262*π/666) = 2*cos(131*π/333) 0.639527849142 = 2*cos(264*π/666) = 2*cos(44*π/111) 0.621621874052 = 2*cos(266*π/666) = 2*cos(133*π/333) 0.603660572314 = 2*cos(268*π/666) = 2*cos(134*π/333) 0.585645542553 = 2*cos(270*π/666) = 2*cos(15*π/37) 0.567578388172 = 2*cos(272*π/666) = 2*cos(136*π/333) 0.549460717215 = 2*cos(274*π/666) = 2*cos(137*π/333) 0.531294142222 = 2*cos(276*π/666) = 2*cos(46*π/111) 0.513080280084 = 2*cos(278*π/666) = 2*cos(139*π/333) 0.494820751904 = 2*cos(280*π/666) = 2*cos(140*π/333) 0.476517182846 = 2*cos(282*π/666) = 2*cos(47*π/111) 0.458171201997 = 2*cos(284*π/666) = 2*cos(142*π/333) 0.439784442215 = 2*cos(286*π/666) = 2*cos(143*π/333) 0.421358539991 = 2*cos(288*π/666) = 2*cos(16*π/37) 0.402895135299 = 2*cos(290*π/666) = 2*cos(145*π/333) 0.384395871449 = 2*cos(292*π/666) = 2*cos(146*π/333) 0.365862394945 = 2*cos(294*π/666) = 2*cos(49*π/111) 0.347296355334 = 2*cos(296*π/666) = 2*cos(4*π/9) 0.328699405063 = 2*cos(298*π/666) = 2*cos(149*π/333) 0.310073199328 = 2*cos(300*π/666) = 2*cos(50*π/111) 0.291419395932 = 2*cos(302*π/666) = 2*cos(151*π/333) 0.272739655132 = 2*cos(304*π/666) = 2*cos(152*π/333) 0.254035639494 = 2*cos(306*π/666) = 2*cos(17*π/37) 0.235309013744 = 2*cos(308*π/666) = 2*cos(154*π/333) 0.216561444621 = 2*cos(310*π/666) = 2*cos(155*π/333) 0.197794600728 = 2*cos(312*π/666) = 2*cos(52*π/111) 0.179010152385 = 2*cos(314*π/666) = 2*cos(157*π/333) 0.160209771476 = 2*cos(316*π/666) = 2*cos(158*π/333) 0.141395131304 = 2*cos(318*π/666) = 2*cos(53*π/111) 0.122567906443 = 2*cos(320*π/666) = 2*cos(160*π/333) 0.103729772584 = 2*cos(322*π/666) = 2*cos(161*π/333) 0.084882406392 = 2*cos(324*π/666) = 2*cos(18*π/37) 0.066027485352 = 2*cos(326*π/666) = 2*cos(163*π/333) 0.047166687622 = 2*cos(328*π/666) = 2*cos(164*π/333) 0.028301691882 = 2*cos(330*π/666) = 2*cos(55*π/111) 0.009434177186 = 2*cos(332*π/666) = 2*cos(166*π/333) Symmetry of Rotations and Cartesian products A1 p+d+f+g+h+i+j+k+l+m z, z², z³, z⁴, z⁵, z⁶ A2 R R_z E1 R+p+d+f+g+h+i+j+k+l+m {R_x, R_y}, {x, y}, {xz, yz}, {xz², yz²}, {xz³, yz³}, {xz⁴, yz⁴}, {xz⁵, yz⁵} E2 d+f+g+h+i+j+k+l+m {x²−y², xy}, {z(x²−y²), xyz}, {z²(x²−y²), xyz²}, {z³(x²−y²), xyz³}, {z⁴(x²−y²), xyz⁴} E3 f+g+h+i+j+k+l+m {x(x²−3y²), y(3x²−y²)}, {xz(x²−3y²), yz(3x²−y²)}, {xz²(x²−3y²), yz²(3x²−y²)}, {xz³(x²−3y²), yz³(3x²−y²)} E4 g+h+i+j+k+l+m {(x²−y²)²−4x²y², xy(x²−y²)}, {z((x²−y²)²−4x²y²), xyz(x²−y²)}, {z²((x²−y²)²−4x²y²), xyz²(x²−y²)} E5 h+i+j+k+l+m {x(x²−(5+2√5)y²)(x²−(5−2√5)y²), y((5+2√5)x²−y²)((5−2√5)x²−y²)}, {xz(x²−(5+2√5)y²)(x²−(5−2√5)y²), yz((5+2√5)x²−y²)((5−2√5)x²−y²)} E6 i+j+k+l+m {x²(x²−3y²)²−y²(3x²−y²)², xy(x²−3y²)(3x²−y²)} E7 j+k+l+m E8 k+l+m E9 l+m E10 m Notes: α The order of the C_666v point group is 1332, and the order of the principal axis (C₆₆₆) is 666. The group has 336 irreducible representations. β The C_666v point group is isomorphic to D_333d, D_333h and D₆₆₆. γ The C_666v point group is generated by two symmetry elements, C₆₆₆ and any σ_v (or, non-canonically, any σ_d). Also, the group may be generated from a σ_v plus a σ_d (some pairs will yield smaller groups, though; choosing a minimum angle is safe). δ There are two different sets of symmetry planes containing the principal axis (z axis in standard orientation). By convention, the set denoted as σ_v has the xz plane as a member, while the yz plane is a member of the σ_d set. ε The lowest nonvanishing multipole moment in C_666v is 2 (dipole moment). ζ This point group is non-Abelian (some symmetry operations are not commutative). Therefore, the character table contains multi-membered classes and degenerate irreducible representations. η Some of the characters in the table are irrational because the order of the principal axis is neither 1,2,3,4 nor 6. These irrational values can be expressed as cosine values, or as solutions of algebraic equations with a leading coefficient of 1. All characters are algebraic integers of a degree much less than half the order of the principal axis. θ The point group corresponds to a polygon inconstructible by the classical means of ruler and compass. Yet it becomes constructible if angle trisection is allowed, e.g., with neusis construction or origami. This is because the order of the principal axis is given by a product of any number of different Pierpont primes (...,5,7,13,17,19,37,73,97,109,163,...) times arbitrary powers of two and three. All characters of this group can be expressed using complex numbers, elementary arithmetic operations, square roots and third roots.

This Character Table for the C_666v point group was created by Gernot Katzer.

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

	C_340v
	C_512v
C₆₆₆	C_666v	C_666h D₆₆₆ D_666h D_666d S₆₆₆

Character table for the C666v point group

Character table for the C_666v point group