D82d E 2 S164 2 C82 2 S164^3 2 C41 2 S164^5 2 C82^3 2 S164^7 2 C41^2 2 S164^9 2 C82^5 2 S164^11 2 C41^3 2 S164^13 2 C82^7 2 S164^15 2 C41^4 2 S164^17 2 C82^9 2 S164^19 2 C41^5 2 S164^21 2 C82^11 2 S164^23 2 C41^6 2 S164^25 2 C82^13 2 S164^27 2 C41^7 2 S164^29 2 C82^15 2 S164^31 2 C41^8 2 S164^33 2 C82^17 2 S164^35 2 C41^9 2 S164^37 2 C82^19 2 S164^39 2 C41^10 2 S4 2 C82^21 2 S164^43 2 C41^11 2 S164^45 2 C82^23 2 S164^47 2 C41^12 2 S164^49 2 C82^25 2 S164^51 2 C41^13 2 S164^53 2 C82^27 2 S164^55 2 C41^14 2 S164^57 2 C82^29 2 S164^59 2 C41^15 2 S164^61 2 C82^31 2 S164^63 2 C41^16 2 S164^65 2 C82^33 2 S164^67 2 C41^17 2 S164^69 2 C82^35 2 S164^71 2 C41^18 2 S164^73 2 C82^37 2 S164^75 2 C41^19 2 S164^77 2 C82^39 2 S164^79 2 C41^20 2 S164^81 C2 82 C2' 82 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 ... ... ....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 ..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 ... ... ..... ....... ......... ........... ............. 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 ... ..T ..... ......T ......... ..........T ............. E1 2.00000 1.99853 1.99413 1.98680 1.97656 1.96342 1.94739 1.92851 1.90679 1.88228 1.85500 1.82501 1.79233 1.75702 1.71914 1.67873 1.63586 1.59059 1.54298 1.49311 1.44104 1.38687 1.33065 1.27248 1.21245 1.15064 1.08714 1.02204 0.95544 0.88744 0.81814 0.74763 0.67603 0.60344 0.52996 0.45571 0.38078 0.30530 0.22937 0.15310 0.07661 0.00000 -0.07661 -0.15310 -0.22937 -0.30530 -0.38078 -0.45571 -0.52996 -0.60344 -0.67603 -0.74763 -0.81814 -0.88744 -0.95544 -1.02204 -1.08714 -1.15064 -1.21245 -1.27248 -1.33065 -1.38687 -1.44104 -1.49311 -1.54298 -1.59059 -1.63586 -1.67873 -1.71914 -1.75702 -1.79233 -1.82501 -1.85500 -1.88228 -1.90679 -1.92851 -1.94739 -1.96342 -1.97656 -1.98680 -1.99413 -1.99853 -2.00000 0.00000 0.00000 ... TT. ..... ....TT. ......... ........TT. ............. 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E71 2.00000 -1.82501 1.33065 -0.60344 -0.22937 1.02204 -1.63586 1.96342 -1.94739 1.59059 -0.95544 0.15310 0.67603 -1.38687 1.85500 -1.99853 1.79233 -1.27248 0.52996 0.30530 -1.08714 1.67873 -1.97656 1.92851 -1.54298 0.88744 -0.07661 -0.74763 1.44104 -1.88228 1.99413 -1.75702 1.21245 -0.45571 -0.38078 1.15064 -1.71914 1.98680 -1.90679 1.49311 -0.81814 0.00000 0.81814 -1.49311 1.90679 -1.98680 1.71914 -1.15064 0.38078 0.45571 -1.21245 1.75702 -1.99413 1.88228 -1.44104 0.74763 0.07661 -0.88744 1.54298 -1.92851 1.97656 -1.67873 1.08714 -0.30530 -0.52996 1.27248 -1.79233 1.99853 -1.85500 1.38687 -0.67603 -0.15310 0.95544 -1.59059 1.94739 -1.96342 1.63586 -1.02204 0.22937 0.60344 -1.33065 1.82501 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E72 2.00000 -1.85500 1.44104 -0.81814 0.07661 0.67603 -1.33065 1.79233 -1.99413 1.90679 -1.54298 0.95544 -0.22937 -0.52996 1.21245 -1.71914 1.97656 -1.94739 1.63586 -1.08714 0.38078 0.38078 -1.08714 1.63586 -1.94739 1.97656 -1.71914 1.21245 -0.52996 -0.22937 0.95544 -1.54298 1.90679 -1.99413 1.79233 -1.33065 0.67603 0.07661 -0.81814 1.44104 -1.85500 2.00000 -1.85500 1.44104 -0.81814 0.07661 0.67603 -1.33065 1.79233 -1.99413 1.90679 -1.54298 0.95544 -0.22937 -0.52996 1.21245 -1.71914 1.97656 -1.94739 1.63586 -1.08714 0.38078 0.38078 -1.08714 1.63586 -1.94739 1.97656 -1.71914 1.21245 -0.52996 -0.22937 0.95544 -1.54298 1.90679 -1.99413 1.79233 -1.33065 0.67603 0.07661 -0.81814 1.44104 -1.85500 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E73 2.00000 -1.88228 1.54298 -1.02204 0.38078 0.30530 -0.95544 1.49311 -1.85500 1.99853 -1.90679 1.59059 -1.08714 0.45571 0.22937 -0.88744 1.44104 -1.82501 1.99413 -1.92851 1.63586 -1.15064 0.52996 0.15310 -0.81814 1.38687 -1.79233 1.98680 -1.94739 1.67873 -1.21245 0.60344 0.07661 -0.74763 1.33065 -1.75702 1.97656 -1.96342 1.71914 -1.27248 0.67603 0.00000 -0.67603 1.27248 -1.71914 1.96342 -1.97656 1.75702 -1.33065 0.74763 -0.07661 -0.60344 1.21245 -1.67873 1.94739 -1.98680 1.79233 -1.38687 0.81814 -0.15310 -0.52996 1.15064 -1.63586 1.92851 -1.99413 1.82501 -1.44104 0.88744 -0.22937 -0.45571 1.08714 -1.59059 1.90679 -1.99853 1.85500 -1.49311 0.95544 -0.30530 -0.38078 1.02204 -1.54298 1.88228 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E74 2.00000 -1.90679 1.63586 -1.21245 0.67603 -0.07661 -0.52996 1.08714 -1.54298 1.85500 -1.99413 1.94739 -1.71914 1.33065 -0.81814 0.22937 0.38078 -0.95544 1.44104 -1.79233 1.97656 -1.97656 1.79233 -1.44104 0.95544 -0.38078 -0.22937 0.81814 -1.33065 1.71914 -1.94739 1.99413 -1.85500 1.54298 -1.08714 0.52996 0.07661 -0.67603 1.21245 -1.63586 1.90679 -2.00000 1.90679 -1.63586 1.21245 -0.67603 0.07661 0.52996 -1.08714 1.54298 -1.85500 1.99413 -1.94739 1.71914 -1.33065 0.81814 -0.22937 -0.38078 0.95544 -1.44104 1.79233 -1.97656 1.97656 -1.79233 1.44104 -0.95544 0.38078 0.22937 -0.81814 1.33065 -1.71914 1.94739 -1.99413 1.85500 -1.54298 1.08714 -0.52996 -0.07661 0.67603 -1.21245 1.63586 -1.90679 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E75 2.00000 -1.92851 1.71914 -1.38687 0.95544 -0.45571 -0.07661 0.60344 -1.08714 1.49311 -1.79233 1.96342 -1.99413 1.88228 -1.63586 1.27248 -0.81814 0.30530 0.22937 -0.74763 1.21245 -1.59059 1.85500 -1.98680 1.97656 -1.82501 1.54298 -1.15064 0.67603 -0.15310 -0.38078 0.88744 -1.33065 1.67873 -1.90679 1.99853 -1.94739 1.75702 -1.44104 1.02204 -0.52996 0.00000 0.52996 -1.02204 1.44104 -1.75702 1.94739 -1.99853 1.90679 -1.67873 1.33065 -0.88744 0.38078 0.15310 -0.67603 1.15064 -1.54298 1.82501 -1.97656 1.98680 -1.85500 1.59059 -1.21245 0.74763 -0.22937 -0.30530 0.81814 -1.27248 1.63586 -1.88228 1.99413 -1.96342 1.79233 -1.49311 1.08714 -0.60344 0.07661 0.45571 -0.95544 1.38687 -1.71914 1.92851 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E76 2.00000 -1.94739 1.79233 -1.54298 1.21245 -0.81814 0.38078 0.07661 -0.52996 0.95544 -1.33065 1.63586 -1.85500 1.97656 -1.99413 1.90679 -1.71914 1.44104 -1.08714 0.67603 -0.22937 -0.22937 0.67603 -1.08714 1.44104 -1.71914 1.90679 -1.99413 1.97656 -1.85500 1.63586 -1.33065 0.95544 -0.52996 0.07661 0.38078 -0.81814 1.21245 -1.54298 1.79233 -1.94739 2.00000 -1.94739 1.79233 -1.54298 1.21245 -0.81814 0.38078 0.07661 -0.52996 0.95544 -1.33065 1.63586 -1.85500 1.97656 -1.99413 1.90679 -1.71914 1.44104 -1.08714 0.67603 -0.22937 -0.22937 0.67603 -1.08714 1.44104 -1.71914 1.90679 -1.99413 1.97656 -1.85500 1.63586 -1.33065 0.95544 -0.52996 0.07661 0.38078 -0.81814 1.21245 -1.54298 1.79233 -1.94739 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ............. E77 2.00000 -1.96342 1.85500 -1.67873 1.44104 -1.15064 0.81814 -0.45571 0.07661 0.30530 -0.67603 1.02204 -1.33065 1.59059 -1.79233 1.92851 -1.99413 1.98680 -1.90679 1.75702 -1.54298 1.27248 -0.95544 0.60344 -0.22937 -0.15310 0.52996 -0.88744 1.21245 -1.49311 1.71914 -1.88228 1.97656 -1.99853 1.94739 -1.82501 1.63586 -1.38687 1.08714 -0.74763 0.38078 0.00000 -0.38078 0.74763 -1.08714 1.38687 -1.63586 1.82501 -1.94739 1.99853 -1.97656 1.88228 -1.71914 1.49311 -1.21245 0.88744 -0.52996 0.15310 0.22937 -0.60344 0.95544 -1.27248 1.54298 -1.75702 1.90679 -1.98680 1.99413 -1.92851 1.79233 -1.59059 1.33065 -1.02204 0.67603 -0.30530 -0.07661 0.45571 -0.81814 1.15064 -1.44104 1.67873 -1.85500 1.96342 -2.00000 0.00000 0.00000 ... ... ..... ....... ......... ........... ..TT......... E78 2.00000 -1.97656 1.90679 -1.79233 1.63586 -1.44104 1.21245 -0.95544 0.67603 -0.38078 0.07661 0.22937 -0.52996 0.81814 -1.08714 1.33065 -1.54298 1.71914 -1.85500 1.94739 -1.99413 1.99413 -1.94739 1.85500 -1.71914 1.54298 -1.33065 1.08714 -0.81814 0.52996 -0.22937 -0.07661 0.38078 -0.67603 0.95544 -1.21245 1.44104 -1.63586 1.79233 -1.90679 1.97656 -2.00000 1.97656 -1.90679 1.79233 -1.63586 1.44104 -1.21245 0.95544 -0.67603 0.38078 -0.07661 -0.22937 0.52996 -0.81814 1.08714 -1.33065 1.54298 -1.71914 1.85500 -1.94739 1.99413 -1.99413 1.94739 -1.85500 1.71914 -1.54298 1.33065 -1.08714 0.81814 -0.52996 0.22937 0.07661 -0.38078 0.67603 -0.95544 1.21245 -1.44104 1.63586 -1.79233 1.90679 -1.97656 2.00000 0.00000 0.00000 ... ... ..... ....... ......... ..TT....... ............. E79 2.00000 -1.98680 1.94739 -1.88228 1.79233 -1.67873 1.54298 -1.38687 1.21245 -1.02204 0.81814 -0.60344 0.38078 -0.15310 -0.07661 0.30530 -0.52996 0.74763 -0.95544 1.15064 -1.33065 1.49311 -1.63586 1.75702 -1.85500 1.92851 -1.97656 1.99853 -1.99413 1.96342 -1.90679 1.82501 -1.71914 1.59059 -1.44104 1.27248 -1.08714 0.88744 -0.67603 0.45571 -0.22937 0.00000 0.22937 -0.45571 0.67603 -0.88744 1.08714 -1.27248 1.44104 -1.59059 1.71914 -1.82501 1.90679 -1.96342 1.99413 -1.99853 1.97656 -1.92851 1.85500 -1.75702 1.63586 -1.49311 1.33065 -1.15064 0.95544 -0.74763 0.52996 -0.30530 0.07661 0.15310 -0.38078 0.60344 -0.81814 1.02204 -1.21245 1.38687 -1.54298 1.67873 -1.79233 1.88228 -1.94739 1.98680 -2.00000 0.00000 0.00000 ... ... ..... ....... ..TT..... ........... ......TT..... E80 2.00000 -1.99413 1.97656 -1.94739 1.90679 -1.85500 1.79233 -1.71914 1.63586 -1.54298 1.44104 -1.33065 1.21245 -1.08714 0.95544 -0.81814 0.67603 -0.52996 0.38078 -0.22937 0.07661 0.07661 -0.22937 0.38078 -0.52996 0.67603 -0.81814 0.95544 -1.08714 1.21245 -1.33065 1.44104 -1.54298 1.63586 -1.71914 1.79233 -1.85500 1.90679 -1.94739 1.97656 -1.99413 2.00000 -1.99413 1.97656 -1.94739 1.90679 -1.85500 1.79233 -1.71914 1.63586 -1.54298 1.44104 -1.33065 1.21245 -1.08714 0.95544 -0.81814 0.67603 -0.52996 0.38078 -0.22937 0.07661 0.07661 -0.22937 0.38078 -0.52996 0.67603 -0.81814 0.95544 -1.08714 1.21245 -1.33065 1.44104 -1.54298 1.63586 -1.71914 1.79233 -1.85500 1.90679 -1.94739 1.97656 -1.99413 2.00000 0.00000 0.00000 ... ... ..... ..TT... ......... ......TT... ............. E81 2.00000 -1.99853 1.99413 -1.98680 1.97656 -1.96342 1.94739 -1.92851 1.90679 -1.88228 1.85500 -1.82501 1.79233 -1.75702 1.71914 -1.67873 1.63586 -1.59059 1.54298 -1.49311 1.44104 -1.38687 1.33065 -1.27248 1.21245 -1.15064 1.08714 -1.02204 0.95544 -0.88744 0.81814 -0.74763 0.67603 -0.60344 0.52996 -0.45571 0.38078 -0.30530 0.22937 -0.15310 0.07661 0.00000 -0.07661 0.15310 -0.22937 0.30530 -0.38078 0.45571 -0.52996 0.60344 -0.67603 0.74763 -0.81814 0.88744 -0.95544 1.02204 -1.08714 1.15064 -1.21245 1.27248 -1.33065 1.38687 -1.44104 1.49311 -1.54298 1.59059 -1.63586 1.67873 -1.71914 1.75702 -1.79233 1.82501 -1.85500 1.88228 -1.90679 1.92851 -1.94739 1.96342 -1.97656 1.98680 -1.99413 1.99853 -2.00000 0.00000 0.00000 TT. ... ..TT. ....... ......TT. ........... ..........TT. Irrational character values: 1.998532362102 = 2*cos(2*π/164) = 2*cos(π/82) 1.994131602367 = 2*cos(4*π/164) = 2*cos(π/41) 1.986804179519 = 2*cos(6*π/164) = 2*cos(3*π/82) 1.976560847561 = 2*cos(8*π/164) = 2*cos(2*π/41) 1.963416639994 = 2*cos(10*π/164) = 2*cos(5*π/82) 1.947390847756 = 2*cos(12*π/164) = 2*cos(3*π/41) 1.928506990906 = 2*cos(14*π/164) = 2*cos(7*π/82) 1.906792784110 = 2*cos(16*π/164) = 2*cos(4*π/41) 1.882280095959 = 2*cos(18*π/164) = 2*cos(9*π/82) 1.855004902204 = 2*cos(20*π/164) = 2*cos(5*π/41) 1.825007232953 = 2*cos(22*π/164) = 2*cos(11*π/82) 1.792331113922 = 2*cos(24*π/164) = 2*cos(6*π/41) 1.757024501822 = 2*cos(26*π/164) = 2*cos(13*π/82) 1.719139213974 = 2*cos(28*π/164) = 2*cos(7*π/41) 1.678730852264 = 2*cos(30*π/164) = 2*cos(15*π/82) 1.635858721533 = 2*cos(32*π/164) = 2*cos(8*π/41) 1.590585742547 = 2*cos(34*π/164) = 2*cos(17*π/82) 1.542978359644 = 2*cos(36*π/164) = 2*cos(9*π/41) 1.493106443224 = 2*cos(38*π/164) = 2*cos(19*π/82) 1.441043187202 = 2*cos(40*π/164) = 2*cos(10*π/41) 1.386865001584 = 2*cos(42*π/164) = 2*cos(21*π/82) 1.330651400331 = 2*cos(44*π/164) = 2*cos(11*π/41) 1.272484884653 = 2*cos(46*π/164) = 2*cos(23*π/82) 1.212450821933 = 2*cos(48*π/164) = 2*cos(12*π/41) 1.150637320437 = 2*cos(50*π/164) = 2*cos(25*π/82) 1.087135100002 = 2*cos(52*π/164) = 2*cos(13*π/41) 1.022037358894 = 2*cos(54*π/164) = 2*cos(27*π/82) 0.955439637025 = 2*cos(56*π/164) = 2*cos(14*π/41) 0.887439675734 = 2*cos(58*π/164) = 2*cos(29*π/82) 0.818137274343 = 2*cos(60*π/164) = 2*cos(15*π/41) 0.747634143682 = 2*cos(62*π/164) = 2*cos(31*π/82) 0.676033756817 = 2*cos(64*π/164) = 2*cos(16*π/41) 0.603441197190 = 2*cos(66*π/164) = 2*cos(33*π/82) 0.529963004393 = 2*cos(68*π/164) = 2*cos(17*π/41) 0.455707017806 = 2*cos(70*π/164) = 2*cos(35*π/82) 0.380782218329 = 2*cos(72*π/164) = 2*cos(18*π/41) 0.305298568438 = 2*cos(74*π/164) = 2*cos(37*π/82) 0.229366850797 = 2*cos(76*π/164) = 2*cos(19*π/41) 0.153098505673 = 2*cos(78*π/164) = 2*cos(39*π/82) 0.076605467380 = 2*cos(80*π/164) = 2*cos(20*π/41) 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 E73 m E74 l E75 k+m E76 j+l E77 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)} E78 h+j+l {z((x2−y2)2−4x2y2), xyz(x2−y2)} E79 g+i+k+m {xz(x2−3y2), yz(3x2−y2)}, {xz3(x2−3y2), yz3(3x2−y2)} E80 f+h+j+l {z(x2−y2), xyz}, {z3(x2−y2), xyz3} E81 R+d+g+i+k+m {Rx, Ry}, {xz, yz}, {xz3, yz3}, {xz5, yz5} Notes: α The order of the D82d point group is 328, and the order of the principal axis (S164) is 164. The group has 85 irreducible representations. β The D82d point group is isomorphic to C164v and D164. γ The D82d point group is generated by two symmetry elements, S164 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 D82d 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. For this group, however, none of the irrational characters can be expressed by a closed algebraic form using real numbers only.
D80d | ||
D81d | ||
C82 C82v C82h D82 D82h | D82d | S82 |
D83d | ||
D84d |
This Character Table for the D82d point group was created by Gernot Katzer.
For other groups and some explanations, see the Main Page.