Character Table for Point Group C255v

Character table for the C_255v point group

C255v E 2 C255 2 C255^2 2 C85 2 C255^4 2 C51 2 C85^2 2 C255^7 2 C255^8 2 C85^3 2 C51^2 2 C255^11 2 C85^4 2 C255^13 2 C255^14 2 C17 2 C255^16 2 C15 2 C85^6 2 C255^19 2 C51^4 2 C85^7 2 C255^22 2 C255^23 2 C85^8 2 C51^5 2 C255^26 2 C85^9 2 C255^28 2 C255^29 2 C17^2 2 C255^31 2 C255^32 2 C85^11 2 C15^2 2 C51^7 2 C85^12 2 C255^37 2 C255^38 2 C85^13 2 C51^8 2 C255^41 2 C85^14 2 C255^43 2 C255^44 2 C17^3 2 C255^46 2 C255^47 2 C85^16 2 C255^49 2 C51^10 2 C5 2 C255^52 2 C255^53 2 C85^18 2 C51^11 2 C255^56 2 C85^19 2 C255^58 2 C255^59 2 C17^4 2 C255^61 2 C255^62 2 C85^21 2 C255^64 2 C51^13 2 C85^22 2 C255^67 2 C15^4 2 C85^23 2 C51^14 2 C255^71 2 C85^24 2 C255^73 2 C255^74 2 C17^5 2 C255^76 2 C255^77 2 C85^26 2 C255^79 2 C51^16 2 C85^27 2 C255^82 2 C255^83 2 C85^28 2 C3 2 C255^86 2 C85^29 2 C255^88 2 C255^89 2 C17^6 2 C255^91 2 C255^92 2 C85^31 2 C255^94 2 C51^19 2 C85^32 2 C255^97 2 C255^98 2 C85^33 2 C51^20 2 C255^101 2 C5^2 2 C255^103 2 C255^104 2 C17^7 2 C255^106 2 C255^107 2 C85^36 2 C255^109 2 C51^22 2 C85^37 2 C255^112 2 C255^113 2 C85^38 2 C51^23 2 C255^116 2 C85^39 2 C255^118 2 C15^7 2 C17^8 2 C255^121 2 C255^122 2 C85^41 2 C255^124 2 C51^25 2 C85^42 2 C255^127 255 sv <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 ... ..T ....T ......T ........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 ..T ... ..... ....... ......... ........... ............. E1 2.00000 1.99939 1.99757 1.99454 1.99029 1.98484 1.97818 1.97032 1.96127 1.95102 1.93959 1.92699 1.91321 1.89827 1.88218 1.86494 1.84658 1.82709 1.80649 1.78480 1.76202 1.73818 1.71328 1.68733 1.66037 1.63239 1.60343 1.57349 1.54260 1.51077 1.47802 1.44437 1.40985 1.37447 1.33826 1.30124 1.26342 1.22484 1.18552 1.14547 1.10473 1.06332 1.02126 0.97859 0.93532 0.89148 0.84710 0.80220 0.75682 0.71098 0.66471 0.61803 0.57098 0.52359 0.47587 0.42787 0.37960 0.33111 0.28241 0.23355 0.18454 0.13542 0.08621 0.03696 -0.01232 -0.06159 -0.11082 -0.15999 -0.20906 -0.25800 -0.30678 -0.35538 -0.40376 -0.45190 -0.49977 -0.54733 -0.59455 -0.64142 -0.68790 -0.73396 -0.77957 -0.82471 -0.86935 -0.91347 -0.95702 -1.00000 -1.04237 -1.08411 -1.12519 -1.16558 -1.20527 -1.24423 -1.28243 -1.31985 -1.35647 -1.39227 -1.42722 -1.46131 -1.49451 -1.52680 -1.55816 -1.58858 -1.61803 -1.64651 -1.67398 -1.70043 -1.72586 -1.75023 -1.77355 -1.79578 -1.81693 -1.83697 -1.85590 -1.87370 -1.89037 -1.90588 -1.92024 -1.93344 -1.94546 -1.95630 -1.96595 -1.97440 -1.98166 -1.98772 -1.99257 -1.99621 -1.99863 -1.99985 0.00000 TT. TT. ..TT. ....TT. ......TT. ........TT. ..........TT. E2 2.00000 1.99757 1.99029 1.97818 1.96127 1.93959 1.91321 1.88218 1.84658 1.80649 1.76202 1.71328 1.66037 1.60343 1.54260 1.47802 1.40985 1.33826 1.26342 1.18552 1.10473 1.02126 0.93532 0.84710 0.75682 0.66471 0.57098 0.47587 0.37960 0.28241 0.18454 0.08621 -0.01232 -0.11082 -0.20906 -0.30678 -0.40376 -0.49977 -0.59455 -0.68790 -0.77957 -0.86935 -0.95702 -1.04237 -1.12519 -1.20527 -1.28243 -1.35647 -1.42722 -1.49451 -1.55816 -1.61803 -1.67398 -1.72586 -1.77355 -1.81693 -1.85590 -1.89037 -1.92024 -1.94546 -1.96595 -1.98166 -1.99257 -1.99863 -1.99985 -1.99621 -1.98772 -1.97440 -1.95630 -1.93344 -1.90588 -1.87370 -1.83697 -1.79578 -1.75023 -1.70043 -1.64651 -1.58858 -1.52680 -1.46131 -1.39227 -1.31985 -1.24423 -1.16558 -1.08411 -1.00000 -0.91347 -0.82471 -0.73396 -0.64142 -0.54733 -0.45190 -0.35538 -0.25800 -0.15999 -0.06159 0.03696 0.13542 0.23355 0.33111 0.42787 0.52359 0.61803 0.71098 0.80220 0.89148 0.97859 1.06332 1.14547 1.22484 1.30124 1.37447 1.44437 1.51077 1.57349 1.63239 1.68733 1.73818 1.78480 1.82709 1.86494 1.89827 1.92699 1.95102 1.97032 1.98484 1.99454 1.99939 0.00000 ... ... TT... ..TT... ....TT... ......TT... ........TT... E3 2.00000 1.99454 1.97818 1.95102 1.91321 1.86494 1.80649 1.73818 1.66037 1.57349 1.47802 1.37447 1.26342 1.14547 1.02126 0.89148 0.75682 0.61803 0.47587 0.33111 0.18454 0.03696 -0.11082 -0.25800 -0.40376 -0.54733 -0.68790 -0.82471 -0.95702 -1.08411 -1.20527 -1.31985 -1.42722 -1.52680 -1.61803 -1.70043 -1.77355 -1.83697 -1.89037 -1.93344 -1.96595 -1.98772 -1.99863 -1.99863 -1.98772 -1.96595 -1.93344 -1.89037 -1.83697 -1.77355 -1.70043 -1.61803 -1.52680 -1.42722 -1.31985 -1.20527 -1.08411 -0.95702 -0.82471 -0.68790 -0.54733 -0.40376 -0.25800 -0.11082 0.03696 0.18454 0.33111 0.47587 0.61803 0.75682 0.89148 1.02126 1.14547 1.26342 1.37447 1.47802 1.57349 1.66037 1.73818 1.80649 1.86494 1.91321 1.95102 1.97818 1.99454 2.00000 1.99454 1.97818 1.95102 1.91321 1.86494 1.80649 1.73818 1.66037 1.57349 1.47802 1.37447 1.26342 1.14547 1.02126 0.89148 0.75682 0.61803 0.47587 0.33111 0.18454 0.03696 -0.11082 -0.25800 -0.40376 -0.54733 -0.68790 -0.82471 -0.95702 -1.08411 -1.20527 -1.31985 -1.42722 -1.52680 -1.61803 -1.70043 -1.77355 -1.83697 -1.89037 -1.93344 -1.96595 -1.98772 -1.99863 0.00000 ... ... ..... TT..... ..TT..... ....TT..... ......TT..... E4 2.00000 1.99029 1.96127 1.91321 1.84658 1.76202 1.66037 1.54260 1.40985 1.26342 1.10473 0.93532 0.75682 0.57098 0.37960 0.18454 -0.01232 -0.20906 -0.40376 -0.59455 -0.77957 -0.95702 -1.12519 -1.28243 -1.42722 -1.55816 -1.67398 -1.77355 -1.85590 -1.92024 -1.96595 -1.99257 -1.99985 -1.98772 -1.95630 -1.90588 -1.83697 -1.75023 -1.64651 -1.52680 -1.39227 -1.24423 -1.08411 -0.91347 -0.73396 -0.54733 -0.35538 -0.15999 0.03696 0.23355 0.42787 0.61803 0.80220 0.97859 1.14547 1.30124 1.44437 1.57349 1.68733 1.78480 1.86494 1.92699 1.97032 1.99454 1.99939 1.98484 1.95102 1.89827 1.82709 1.73818 1.63239 1.51077 1.37447 1.22484 1.06332 0.89148 0.71098 0.52359 0.33111 0.13542 -0.06159 -0.25800 -0.45190 -0.64142 -0.82471 -1.00000 -1.16558 -1.31985 -1.46131 -1.58858 -1.70043 -1.79578 -1.87370 -1.93344 -1.97440 -1.99621 -1.99863 -1.98166 -1.94546 -1.89037 -1.81693 -1.72586 -1.61803 -1.49451 -1.35647 -1.20527 -1.04237 -0.86935 -0.68790 -0.49977 -0.30678 -0.11082 0.08621 0.28241 0.47587 0.66471 0.84710 1.02126 1.18552 1.33826 1.47802 1.60343 1.71328 1.80649 1.88218 1.93959 1.97818 1.99757 0.00000 ... ... ..... ....... TT....... ..TT....... ....TT....... E5 2.00000 1.98484 1.93959 1.86494 1.76202 1.63239 1.47802 1.30124 1.10473 0.89148 0.66471 0.42787 0.18454 -0.06159 -0.30678 -0.54733 -0.77957 -1.00000 -1.20527 -1.39227 -1.55816 -1.70043 -1.81693 -1.90588 -1.96595 -1.99621 -1.99621 -1.96595 -1.90588 -1.81693 -1.70043 -1.55816 -1.39227 -1.20527 -1.00000 -0.77957 -0.54733 -0.30678 -0.06159 0.18454 0.42787 0.66471 0.89148 1.10473 1.30124 1.47802 1.63239 1.76202 1.86494 1.93959 1.98484 2.00000 1.98484 1.93959 1.86494 1.76202 1.63239 1.47802 1.30124 1.10473 0.89148 0.66471 0.42787 0.18454 -0.06159 -0.30678 -0.54733 -0.77957 -1.00000 -1.20527 -1.39227 -1.55816 -1.70043 -1.81693 -1.90588 -1.96595 -1.99621 -1.99621 -1.96595 -1.90588 -1.81693 -1.70043 -1.55816 -1.39227 -1.20527 -1.00000 -0.77957 -0.54733 -0.30678 -0.06159 0.18454 0.42787 0.66471 0.89148 1.10473 1.30124 1.47802 1.63239 1.76202 1.86494 1.93959 1.98484 2.00000 1.98484 1.93959 1.86494 1.76202 1.63239 1.47802 1.30124 1.10473 0.89148 0.66471 0.42787 0.18454 -0.06159 -0.30678 -0.54733 -0.77957 -1.00000 -1.20527 -1.39227 -1.55816 -1.70043 -1.81693 -1.90588 -1.96595 -1.99621 0.00000 ... ... ..... ....... ......... TT......... ..TT......... E6 2.00000 1.97818 1.91321 1.80649 1.66037 1.47802 1.26342 1.02126 0.75682 0.47587 0.18454 -0.11082 -0.40376 -0.68790 -0.95702 -1.20527 -1.42722 -1.61803 -1.77355 -1.89037 -1.96595 -1.99863 -1.98772 -1.93344 -1.83697 -1.70043 -1.52680 -1.31985 -1.08411 -0.82471 -0.54733 -0.25800 0.03696 0.33111 0.61803 0.89148 1.14547 1.37447 1.57349 1.73818 1.86494 1.95102 1.99454 1.99454 1.95102 1.86494 1.73818 1.57349 1.37447 1.14547 0.89148 0.61803 0.33111 0.03696 -0.25800 -0.54733 -0.82471 -1.08411 -1.31985 -1.52680 -1.70043 -1.83697 -1.93344 -1.98772 -1.99863 -1.96595 -1.89037 -1.77355 -1.61803 -1.42722 -1.20527 -0.95702 -0.68790 -0.40376 -0.11082 0.18454 0.47587 0.75682 1.02126 1.26342 1.47802 1.66037 1.80649 1.91321 1.97818 2.00000 1.97818 1.91321 1.80649 1.66037 1.47802 1.26342 1.02126 0.75682 0.47587 0.18454 -0.11082 -0.40376 -0.68790 -0.95702 -1.20527 -1.42722 -1.61803 -1.77355 -1.89037 -1.96595 -1.99863 -1.98772 -1.93344 -1.83697 -1.70043 -1.52680 -1.31985 -1.08411 -0.82471 -0.54733 -0.25800 0.03696 0.33111 0.61803 0.89148 1.14547 1.37447 1.57349 1.73818 1.86494 1.95102 1.99454 0.00000 ... ... ..... ....... ......... ........... TT........... E7 2.00000 1.97032 1.88218 1.73818 1.54260 1.30124 1.02126 0.71098 0.37960 0.03696 -0.30678 -0.64142 -0.95702 -1.24423 -1.49451 -1.70043 -1.85590 -1.95630 -1.99863 -1.98166 -1.90588 -1.77355 -1.58858 -1.35647 -1.08411 -0.77957 -0.45190 -0.11082 0.23355 0.57098 0.89148 1.18552 1.44437 1.66037 1.82709 1.93959 1.99454 1.99029 1.92699 1.80649 1.63239 1.40985 1.14547 0.84710 0.52359 0.18454 -0.15999 -0.49977 -0.82471 -1.12519 -1.39227 -1.61803 -1.79578 -1.92024 -1.98772 -1.99621 -1.94546 -1.83697 -1.67398 -1.46131 -1.20527 -0.91347 -0.59455 -0.25800 0.08621 0.42787 0.75682 1.06332 1.33826 1.57349 1.76202 1.89827 1.97818 1.99939 1.96127 1.86494 1.71328 1.51077 1.26342 0.97859 0.66471 0.33111 -0.01232 -0.35538 -0.68790 -1.00000 -1.28243 -1.52680 -1.72586 -1.87370 -1.96595 -1.99985 -1.97440 -1.89037 -1.75023 -1.55816 -1.31985 -1.04237 -0.73396 -0.40376 -0.06159 0.28241 0.61803 0.93532 1.22484 1.47802 1.68733 1.84658 1.95102 1.99757 1.98484 1.91321 1.78480 1.60343 1.37447 1.10473 0.80220 0.47587 0.13542 -0.20906 -0.54733 -0.86935 -1.16558 -1.42722 -1.64651 -1.81693 -1.93344 -1.99257 0.00000 ... ... ..... ....... ......... ........... ............. E8 2.00000 1.96127 1.84658 1.66037 1.40985 1.10473 0.75682 0.37960 -0.01232 -0.40376 -0.77957 -1.12519 -1.42722 -1.67398 -1.85590 -1.96595 -1.99985 -1.95630 -1.83697 -1.64651 -1.39227 -1.08411 -0.73396 -0.35538 0.03696 0.42787 0.80220 1.14547 1.44437 1.68733 1.86494 1.97032 1.99939 1.95102 1.82709 1.63239 1.37447 1.06332 0.71098 0.33111 -0.06159 -0.45190 -0.82471 -1.16558 -1.46131 -1.70043 -1.87370 -1.97440 -1.99863 -1.94546 -1.81693 -1.61803 -1.35647 -1.04237 -0.68790 -0.30678 0.08621 0.47587 0.84710 1.18552 1.47802 1.71328 1.88218 1.97818 1.99757 1.93959 1.80649 1.60343 1.33826 1.02126 0.66471 0.28241 -0.11082 -0.49977 -0.86935 -1.20527 -1.49451 -1.72586 -1.89037 -1.98166 -1.99621 -1.93344 -1.79578 -1.58858 -1.31985 -1.00000 -0.64142 -0.25800 0.13542 0.52359 0.89148 1.22484 1.51077 1.73818 1.89827 1.98484 1.99454 1.92699 1.78480 1.57349 1.30124 0.97859 0.61803 0.23355 -0.15999 -0.54733 -0.91347 -1.24423 -1.52680 -1.75023 -1.90588 -1.98772 -1.99257 -1.92024 -1.77355 -1.55816 -1.28243 -0.95702 -0.59455 -0.20906 0.18454 0.57098 0.93532 1.26342 1.54260 1.76202 1.91321 1.99029 0.00000 ... ... ..... ....... ......... ........... ............. 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E125 2.00000 -1.99621 1.98484 -1.96595 1.93959 -1.90588 1.86494 -1.81693 1.76202 -1.70043 1.63239 -1.55816 1.47802 -1.39227 1.30124 -1.20527 1.10473 -1.00000 0.89148 -0.77957 0.66471 -0.54733 0.42787 -0.30678 0.18454 -0.06159 -0.06159 0.18454 -0.30678 0.42787 -0.54733 0.66471 -0.77957 0.89148 -1.00000 1.10473 -1.20527 1.30124 -1.39227 1.47802 -1.55816 1.63239 -1.70043 1.76202 -1.81693 1.86494 -1.90588 1.93959 -1.96595 1.98484 -1.99621 2.00000 -1.99621 1.98484 -1.96595 1.93959 -1.90588 1.86494 -1.81693 1.76202 -1.70043 1.63239 -1.55816 1.47802 -1.39227 1.30124 -1.20527 1.10473 -1.00000 0.89148 -0.77957 0.66471 -0.54733 0.42787 -0.30678 0.18454 -0.06159 -0.06159 0.18454 -0.30678 0.42787 -0.54733 0.66471 -0.77957 0.89148 -1.00000 1.10473 -1.20527 1.30124 -1.39227 1.47802 -1.55816 1.63239 -1.70043 1.76202 -1.81693 1.86494 -1.90588 1.93959 -1.96595 1.98484 -1.99621 2.00000 -1.99621 1.98484 -1.96595 1.93959 -1.90588 1.86494 -1.81693 1.76202 -1.70043 1.63239 -1.55816 1.47802 -1.39227 1.30124 -1.20527 1.10473 -1.00000 0.89148 -0.77957 0.66471 -0.54733 0.42787 -0.30678 0.18454 -0.06159 0.00000 ... ... ..... ....... ......... ........... ............. 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Irrational character values: 1.999392903956 = 2*cos(2*π/255) 1.997571984389 = 2*cos(4*π/255) 1.994538346772 = 2*cos(6*π/255) = 2*cos(2*π/85) 1.990293832814 = 2*cos(8*π/255) 1.984841019344 = 2*cos(10*π/255) = 2*cos(2*π/51) = (1+√17+√34+2*√17−2*√17−3*√17+√34+2*√17−2*√34−2*√17)/16 + √6*√17+√17−√34+2*√17+2*√17−3*√17−√34+2*√17+2*√34−2*√17/8 1.978183216742 = 2*cos(12*π/255) = 2*cos(4*π/85) 1.970324466935 = 2*cos(14*π/255) 1.961269540938 = 2*cos(16*π/255) 1.951023935961 = 2*cos(18*π/255) = 2*cos(6*π/85) 1.939593872070 = 2*cos(20*π/255) = 2*cos(4*π/51) = (1+√17−√34+2*√17+2*√17−3*√17−√34+2*√17+2*√34−2*√17)/16 + √6*√17+√17+√34+2*√17+2*√17−3*√17+√34+2*√17−2*√34−2*√17/8 1.926986288412 = 2*cos(22*π/255) 1.913208839001 = 2*cos(24*π/255) = 2*cos(8*π/85) 1.898269888072 = 2*cos(26*π/255) 1.882178505003 = 2*cos(28*π/255) 1.864944458809 = 2*cos(30*π/255) = 2*cos(2*π/17) = (−1+√17+√34−2*√17+2*√17+3*√17−√34−2*√17−2*√34+2*√17)/8 1.846578212211 = 2*cos(32*π/255) 1.827090915285 = 2*cos(34*π/255) = 2*cos(2*π/15) = (1+√5+√30−6*√5)/4 1.806494398692 = 2*cos(36*π/255) = 2*cos(12*π/85) 1.784801166496 = 2*cos(38*π/255) 1.762024388572 = 2*cos(40*π/255) = 2*cos(8*π/51) = (1+√17+√34+2*√17+2*√17−3*√17+√34+2*√17−2*√34−2*√17)/16 + √6*√17+√17−√34+2*√17−2*√17−3*√17−√34+2*√17+2*√34−2*√17/8 1.738177892611 = 2*cos(42*π/255) = 2*cos(14*π/85) 1.713276155728 = 2*cos(44*π/255) 1.687334295668 = 2*cos(46*π/255) 1.660368061631 = 2*cos(48*π/255) = 2*cos(16*π/85) 1.632393824712 = 2*cos(50*π/255) = 2*cos(10*π/51) = (1−√17−√34−2*√17+2*√17+3*√17−√34−2*√17−2*√34+2*√17)/16 + √6*√17−√17+√34−2*√17+2*√17+3*√17+√34−2*√17+2*√34+2*√17/8 1.603428567960 = 2*cos(52*π/255) 1.573489876067 = 2*cos(54*π/255) = 2*cos(18*π/85) 1.542595924694 = 2*cos(56*π/255) 1.510765469438 = 2*cos(58*π/255) 1.478017834441 = 2*cos(60*π/255) = 2*cos(4*π/17) = (−1+√17−√34−2*√17+2*√17+3*√17+√34−2*√17+2*√34+2*√17)/8 1.444372900664 = 2*cos(62*π/255) 1.409851093812 = 2*cos(64*π/255) 1.374473371939 = 2*cos(66*π/255) = 2*cos(22*π/85) 1.338261212718 = 2*cos(68*π/255) = 2*cos(4*π/15) = (1−√5+√30+6*√5)/4 1.301236600408 = 2*cos(70*π/255) = 2*cos(14*π/51) = (1−√17+√34−2*√17+2*√17+3*√17+√34−2*√17+2*√34+2*√17)/16 + √6*√17−√17−√34−2*√17−2*√17+3*√17−√34−2*√17−2*√34+2*√17/8 1.263422012507 = 2*cos(72*π/255) = 2*cos(24*π/85) 1.224840406099 = 2*cos(74*π/255) 1.185515203925 = 2*cos(76*π/255) 1.145470280161 = 2*cos(78*π/255) = 2*cos(26*π/85) 1.104729945921 = 2*cos(80*π/255) = 2*cos(16*π/51) = (1+√17−√34+2*√17−2*√17−3*√17−√34+2*√17+2*√34−2*√17)/16 + √6*√17+√17+√34+2*√17−2*√17−3*√17+√34+2*√17−2*√34−2*√17/8 1.063318934501 = 2*cos(82*π/255) 1.021262386362 = 2*cos(84*π/255) = 2*cos(28*π/85) 0.978585833868 = 2*cos(86*π/255) 0.935315185785 = 2*cos(88*π/255) 0.891476711553 = 2*cos(90*π/255) = 2*cos(6*π/17) = (−1−√17+√34+2*√17+2*√17−3*√17−√34+2*√17+2*√34−2*√17)/8 0.847097025336 = 2*cos(92*π/255) 0.802203069865 = 2*cos(94*π/255) 0.756822100085 = 2*cos(96*π/255) = 2*cos(32*π/85) 0.710981666601 = 2*cos(98*π/255) 0.664709598959 = 2*cos(100*π/255) = 2*cos(20*π/51) = (1−√17+√34−2*√17+2*√17+3*√17+√34−2*√17+2*√34+2*√17)/16 − √6*√17−√17−√34−2*√17−2*√17+3*√17−√34−2*√17−2*√34+2*√17/8 0.618033988750 = 2*cos(102*π/255) = 2*cos(2*π/5) = (√5−1)/2 0.570983172551 = 2*cos(104*π/255) 0.523585714726 = 2*cos(106*π/255) 0.475870390085 = 2*cos(108*π/255) = 2*cos(36*π/85) 0.427866166413 = 2*cos(110*π/255) = 2*cos(22*π/51) = (1−√17+√34−2*√17−2*√17+3*√17+√34−2*√17+2*√34+2*√17)/16 + √6*√17−√17−√34−2*√17+2*√17+3*√17−√34−2*√17−2*√34+2*√17/8 0.379602186884 = 2*cos(112*π/255) 0.331107752368 = 2*cos(114*π/255) = 2*cos(38*π/85) 0.282412303646 = 2*cos(116*π/255) 0.233545403532 = 2*cos(118*π/255) 0.184536718927 = 2*cos(120*π/255) = 2*cos(8*π/17) = (−1+√17+√34−2*√17−2*√17+3*√17−√34−2*√17−2*√34+2*√17)/8 0.135416002809 = 2*cos(122*π/255) 0.086213076173 = 2*cos(124*π/255) 0.036957809918 = 2*cos(126*π/255) = 2*cos(42*π/85) -0.012319893276 = 2*cos(128*π/255) -0.061590117112 = 2*cos(130*π/255) = 2*cos(26*π/51) = −((−1−√17−√34+2*√17−2*√17−3*√17+√34+2*√17−2*√34−2*√17)/16 + √6*√17+√17−√34+2*√17−2*√17−3*√17−√34+2*√17+2*√34−2*√17/8) -0.110822949832 = 2*cos(132*π/255) = 2*cos(44*π/85) -0.159988502377 = 2*cos(134*π/255) -0.209056926535 = 2*cos(136*π/255) = 2*cos(8*π/15) = −(√30−6*√5−√5−1)/4 -0.257998433060 = 2*cos(138*π/255) = 2*cos(46*π/85) -0.306783309757 = 2*cos(140*π/255) = 2*cos(28*π/51) = −((−1+√17+√34−2*√17+2*√17+3*√17−√34−2*√17−2*√34+2*√17)/16 − √6*√17−√17+√34−2*√17−2*√17+3*√17+√34−2*√17+2*√34+2*√17/8) -0.355381939521 = 2*cos(142*π/255) -0.403764818314 = 2*cos(144*π/255) = 2*cos(48*π/85) -0.451902573083 = 2*cos(146*π/255) -0.499765979588 = 2*cos(148*π/255) -0.547325980144 = 2*cos(150*π/255) = 2*cos(10*π/17) = −(1+√17−√34+2*√17+2*√17−3*√17−√34+2*√17+2*√34−2*√17)/8 -0.594553701262 = 2*cos(152*π/255) -0.641420471181 = 2*cos(154*π/255) -0.687897837268 = 2*cos(156*π/255) = 2*cos(52*π/85) -0.733957583299 = 2*cos(158*π/255) -0.779571746585 = 2*cos(160*π/255) = 2*cos(32*π/51) = −((−1−√17−√34+2*√17+2*√17−3*√17+√34+2*√17−2*√34−2*√17)/16 + √6*√17+√17−√34+2*√17+2*√17−3*√17−√34+2*√17+2*√34−2*√17/8) -0.824712634948 = 2*cos(162*π/255) = 2*cos(54*π/85) -0.869352843532 = 2*cos(164*π/255) -0.913465271444 = 2*cos(166*π/255) -0.957023138203 = 2*cos(168*π/255) = 2*cos(56*π/85) -1.042369765753 = 2*cos(172*π/255) -1.084106712945 = 2*cos(174*π/255) = 2*cos(58*π/85) -1.125185503240 = 2*cos(176*π/255) -1.165581197866 = 2*cos(178*π/255) -1.205269272759 = 2*cos(180*π/255) = 2*cos(12*π/17) = −(1+√17+√34+2*√17−2*√17−3*√17+√34+2*√17−2*√34−2*√17)/8 -1.244225633443 = 2*cos(182*π/255) -1.282426629667 = 2*cos(184*π/255) -1.319849069757 = 2*cos(186*π/255) = 2*cos(62*π/85) -1.356470234698 = 2*cos(188*π/255) -1.392267891926 = 2*cos(190*π/255) = 2*cos(38*π/51) = −((−1−√17+√34+2*√17−2*√17−3*√17−√34+2*√17+2*√34−2*√17)/16 + √6*√17+√17+√34+2*√17+2*√17−3*√17+√34+2*√17−2*√34−2*√17/8) -1.427220308824 = 2*cos(192*π/255) = 2*cos(64*π/85) -1.461306265917 = 2*cos(194*π/255) -1.494505069758 = 2*cos(196*π/255) -1.526796565482 = 2*cos(198*π/255) = 2*cos(66*π/85) -1.558161149051 = 2*cos(200*π/255) = 2*cos(40*π/51) = −((−1+√17+√34−2*√17+2*√17+3*√17−√34−2*√17−2*√34+2*√17)/16 + √6*√17−√17+√34−2*√17−2*√17+3*√17+√34−2*√17+2*√34+2*√17/8) -1.588579779151 = 2*cos(202*π/255) -1.618033988750 = 2*cos(204*π/255) = 2*cos(4*π/5) = −(√5+1)/2 -1.646505896315 = 2*cos(206*π/255) -1.673978216664 = 2*cos(208*π/255) -1.700434271459 = 2*cos(210*π/255) = 2*cos(14*π/17) = −(1+√17+√34+2*√17+2*√17−3*√17+√34+2*√17−2*√34−2*√17)/8 -1.725857999335 = 2*cos(212*π/255) -1.750233965646 = 2*cos(214*π/255) -1.773547371840 = 2*cos(216*π/255) = 2*cos(72*π/85) -1.795784064441 = 2*cos(218*π/255) -1.816930543639 = 2*cos(220*π/255) = 2*cos(44*π/51) = −((−1+√17+√34−2*√17−2*√17+3*√17−√34−2*√17−2*√34+2*√17)/16 + √6*√17−√17+√34−2*√17+2*√17+3*√17+√34−2*√17+2*√34+2*√17/8) -1.836973971492 = 2*cos(222*π/255) = 2*cos(74*π/85) -1.855902179713 = 2*cos(224*π/255) -1.873703677063 = 2*cos(226*π/255) -1.890367656322 = 2*cos(228*π/255) = 2*cos(76*π/85) -1.905884000854 = 2*cos(230*π/255) = 2*cos(46*π/51) = −((−1+√17−√34−2*√17+2*√17+3*√17+√34−2*√17+2*√34+2*√17)/16 + √6*√17−√17−√34−2*√17+2*√17+3*√17−√34−2*√17−2*√34+2*√17/8) -1.920243290749 = 2*cos(232*π/255) -1.933436808538 = 2*cos(234*π/255) = 2*cos(78*π/85) -1.945456544489 = 2*cos(236*π/255) -1.956295201468 = 2*cos(238*π/255) = 2*cos(14*π/15) = −(√5−1+√30+6*√5)/4 -1.965946199368 = 2*cos(240*π/255) = 2*cos(16*π/17) = −(1−√17+√34−2*√17+2*√17+3*√17+√34−2*√17+2*√34+2*√17)/8 -1.974403679107 = 2*cos(242*π/255) -1.981662506183 = 2*cos(244*π/255) -1.987718273791 = 2*cos(246*π/255) = 2*cos(82*π/85) -1.992567305497 = 2*cos(248*π/255) -1.996206657474 = 2*cos(250*π/255) = 2*cos(50*π/51) = −((−1−√17+√34+2*√17+2*√17−3*√17−√34+2*√17+2*√34−2*√17)/16 + √6*√17+√17+√34+2*√17−2*√17−3*√17+√34+2*√17−2*√34−2*√17/8) -1.998634120286 = 2*cos(252*π/255) = 2*cos(84*π/85) -1.999848220230 = 2*cos(254*π/255) 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_255v point group is 510, and the order of the principal axis (C₂₅₅) is 255. The group has 129 irreducible representations. β The C_255v point group is isomorphic to D₂₅₅. γ The C_255v point group is generated by two symmetry elements, C₂₅₅ and any σ_v. Also, the group may be generated from two σ_v planes (some pairs will yield smaller groups, though; choosing a minimum angle is safe). δ The lowest nonvanishing multipole moment in C_255v is 2 (dipole moment). ε This point group is non-Abelian (some symmetry operations are not commutative). Therefore, the character table contains multi-membered classes and degenerate irreducible representations. ζ Some of the characters in the table are irrational because the order of the principal axis is neither 1,2,3,4 nor 6. These irrational values can be expressed as cosine values, or as solutions of algebraic equations with a leading coefficient of 1. All characters are algebraic integers of a degree much less than half the order of the principal axis. η The point group corresponds to a constructible polygon, as the order of the principal axis is a product of any number of different Fermat primes (3,5,17,257,65537) times an arbitrary power of two. Therefore, all characters have an algebraic degree which is a power of two and can be expressed as radicals involving only square roots and integer numbers. Some of these algebraic expressions, however, are too complicated to be included in the list above. θ 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. ι 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 255-gon can be derived from the regular 17-gon by overlaying it with both a equilateral triangle and a regular pentagon.

This Character Table for the C_255v point group was created by Gernot Katzer.

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

	C_204v
	C_240v
C₂₅₅	C_255v	C_255h D₂₅₅ D_255h D_255d
	C_256v
	C_257v

Character table for the C255v point group

Character table for the C_255v point group