Character Table for Point Group D51d

Character table for the D_51d point group

D51d E 2 C51 2 C51^2 2 C17 2 C51^4 2 C51^5 2 C17^2 2 C51^7 2 C51^8 2 C17^3 2 C51^10 2 C51^11 2 C17^4 2 C51^13 2 C51^14 2 C17^5 2 C51^16 2 C3 2 C17^6 2 C51^19 2 C51^20 2 C17^7 2 C51^22 2 C51^23 2 C17^8 2 C51^25 51 C2' i 2 S102 2 S34 2 S102^5 2 S102^7 2 S34^3 2 S102^11 2 S102^13 2 S34^5 2 S6 2 S102^19 2 S34^7 2 S102^23 2 S102^25 2 S34^9 2 S102^29 2 S102^31 2 S34^11 2 S102^35 2 S102^37 2 S34^13 2 S102^41 2 S102^43 2 S34^15 2 S102^47 2 S102^49 51 sd <R> <p> <—d—> <——f——> <———g———> <————h————> <—————i—————> A1g 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 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 A2g 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 -1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 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 ... ..... ....... ......... ........... ............. E1g 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 2.00000 -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 0.00000 TT. ... ..TT. ....... ......TT. ........... ..........TT. E2g 2.00000 1.93959 1.76202 1.47802 1.10473 0.66471 0.18454 -0.30678 -0.77957 -1.20527 -1.55816 -1.81693 -1.96595 -1.99621 -1.90588 -1.70043 -1.39227 -1.00000 -0.54733 -0.06159 0.42787 0.89148 1.30124 1.63239 1.86494 1.98484 0.00000 2.00000 1.98484 1.86494 1.63239 1.30124 0.89148 0.42787 -0.06159 -0.54733 -1.00000 -1.39227 -1.70043 -1.90588 -1.99621 -1.96595 -1.81693 -1.55816 -1.20527 -0.77957 -0.30678 0.18454 0.66471 1.10473 1.47802 1.76202 1.93959 0.00000 ... ... TT... ....... ....TT... ........... ........TT... E3g 2.00000 1.86494 1.47802 0.89148 0.18454 -0.54733 -1.20527 -1.70043 -1.96595 -1.96595 -1.70043 -1.20527 -0.54733 0.18454 0.89148 1.47802 1.86494 2.00000 1.86494 1.47802 0.89148 0.18454 -0.54733 -1.20527 -1.70043 -1.96595 0.00000 2.00000 -1.96595 -1.70043 -1.20527 -0.54733 0.18454 0.89148 1.47802 1.86494 2.00000 1.86494 1.47802 0.89148 0.18454 -0.54733 -1.20527 -1.70043 -1.96595 -1.96595 -1.70043 -1.20527 -0.54733 0.18454 0.89148 1.47802 1.86494 0.00000 ... ... ..... ....... ..TT..... ........... ......TT..... E4g 2.00000 1.76202 1.10473 0.18454 -0.77957 -1.55816 -1.96595 -1.90588 -1.39227 -0.54733 0.42787 1.30124 1.86494 1.98484 1.63239 0.89148 -0.06159 -1.00000 -1.70043 -1.99621 -1.81693 -1.20527 -0.30678 0.66471 1.47802 1.93959 0.00000 2.00000 1.93959 1.47802 0.66471 -0.30678 -1.20527 -1.81693 -1.99621 -1.70043 -1.00000 -0.06159 0.89148 1.63239 1.98484 1.86494 1.30124 0.42787 -0.54733 -1.39227 -1.90588 -1.96595 -1.55816 -0.77957 0.18454 1.10473 1.76202 0.00000 ... ... ..... ....... TT....... ........... ....TT....... E5g 2.00000 1.63239 0.66471 -0.54733 -1.55816 -1.99621 -1.70043 -0.77957 0.42787 1.47802 1.98484 1.76202 0.89148 -0.30678 -1.39227 -1.96595 -1.81693 -1.00000 0.18454 1.30124 1.93959 1.86494 1.10473 -0.06159 -1.20527 -1.90588 0.00000 2.00000 -1.90588 -1.20527 -0.06159 1.10473 1.86494 1.93959 1.30124 0.18454 -1.00000 -1.81693 -1.96595 -1.39227 -0.30678 0.89148 1.76202 1.98484 1.47802 0.42787 -0.77957 -1.70043 -1.99621 -1.55816 -0.54733 0.66471 1.63239 0.00000 ... ... ..... ....... ......... ........... ..TT......... E6g 2.00000 1.47802 0.18454 -1.20527 -1.96595 -1.70043 -0.54733 0.89148 1.86494 1.86494 0.89148 -0.54733 -1.70043 -1.96595 -1.20527 0.18454 1.47802 2.00000 1.47802 0.18454 -1.20527 -1.96595 -1.70043 -0.54733 0.89148 1.86494 0.00000 2.00000 1.86494 0.89148 -0.54733 -1.70043 -1.96595 -1.20527 0.18454 1.47802 2.00000 1.47802 0.18454 -1.20527 -1.96595 -1.70043 -0.54733 0.89148 1.86494 1.86494 0.89148 -0.54733 -1.70043 -1.96595 -1.20527 0.18454 1.47802 0.00000 ... ... ..... ....... ......... ........... TT........... E7g 2.00000 1.30124 -0.30678 -1.70043 -1.90588 -0.77957 0.89148 1.93959 1.63239 0.18454 -1.39227 -1.99621 -1.20527 0.42787 1.76202 1.86494 0.66471 -1.00000 -1.96595 -1.55816 -0.06159 1.47802 1.98484 1.10473 -0.54733 -1.81693 0.00000 2.00000 -1.81693 -0.54733 1.10473 1.98484 1.47802 -0.06159 -1.55816 -1.96595 -1.00000 0.66471 1.86494 1.76202 0.42787 -1.20527 -1.99621 -1.39227 0.18454 1.63239 1.93959 0.89148 -0.77957 -1.90588 -1.70043 -0.30678 1.30124 0.00000 ... ... ..... ....... ......... ........... ............. E8g 2.00000 1.10473 -0.77957 -1.96595 -1.39227 0.42787 1.86494 1.63239 -0.06159 -1.70043 -1.81693 -0.30678 1.47802 1.93959 0.66471 -1.20527 -1.99621 -1.00000 0.89148 1.98484 1.30124 -0.54733 -1.90588 -1.55816 0.18454 1.76202 0.00000 2.00000 1.76202 0.18454 -1.55816 -1.90588 -0.54733 1.30124 1.98484 0.89148 -1.00000 -1.99621 -1.20527 0.66471 1.93959 1.47802 -0.30678 -1.81693 -1.70043 -0.06159 1.63239 1.86494 0.42787 -1.39227 -1.96595 -0.77957 1.10473 0.00000 ... ... ..... ....... ......... ........... ............. E9g 2.00000 0.89148 -1.20527 -1.96595 -0.54733 1.47802 1.86494 0.18454 -1.70043 -1.70043 0.18454 1.86494 1.47802 -0.54733 -1.96595 -1.20527 0.89148 2.00000 0.89148 -1.20527 -1.96595 -0.54733 1.47802 1.86494 0.18454 -1.70043 0.00000 2.00000 -1.70043 0.18454 1.86494 1.47802 -0.54733 -1.96595 -1.20527 0.89148 2.00000 0.89148 -1.20527 -1.96595 -0.54733 1.47802 1.86494 0.18454 -1.70043 -1.70043 0.18454 1.86494 1.47802 -0.54733 -1.96595 -1.20527 0.89148 0.00000 ... ... ..... ....... ......... ........... ............. E10g 2.00000 0.66471 -1.55816 -1.70043 0.42787 1.98484 0.89148 -1.39227 -1.81693 0.18454 1.93959 1.10473 -1.20527 -1.90588 -0.06159 1.86494 1.30124 -1.00000 -1.96595 -0.30678 1.76202 1.47802 -0.77957 -1.99621 -0.54733 1.63239 0.00000 2.00000 1.63239 -0.54733 -1.99621 -0.77957 1.47802 1.76202 -0.30678 -1.96595 -1.00000 1.30124 1.86494 -0.06159 -1.90588 -1.20527 1.10473 1.93959 0.18454 -1.81693 -1.39227 0.89148 1.98484 0.42787 -1.70043 -1.55816 0.66471 0.00000 ... ... ..... ....... ......... ........... ............. E11g 2.00000 0.42787 -1.81693 -1.20527 1.30124 1.76202 -0.54733 -1.99621 -0.30678 1.86494 1.10473 -1.39227 -1.70043 0.66471 1.98484 0.18454 -1.90588 -1.00000 1.47802 1.63239 -0.77957 -1.96595 -0.06159 1.93959 0.89148 -1.55816 0.00000 2.00000 -1.55816 0.89148 1.93959 -0.06159 -1.96595 -0.77957 1.63239 1.47802 -1.00000 -1.90588 0.18454 1.98484 0.66471 -1.70043 -1.39227 1.10473 1.86494 -0.30678 -1.99621 -0.54733 1.76202 1.30124 -1.20527 -1.81693 0.42787 0.00000 ... ... ..... ....... ......... ........... ............. E12g 2.00000 0.18454 -1.96595 -0.54733 1.86494 0.89148 -1.70043 -1.20527 1.47802 1.47802 -1.20527 -1.70043 0.89148 1.86494 -0.54733 -1.96595 0.18454 2.00000 0.18454 -1.96595 -0.54733 1.86494 0.89148 -1.70043 -1.20527 1.47802 0.00000 2.00000 1.47802 -1.20527 -1.70043 0.89148 1.86494 -0.54733 -1.96595 0.18454 2.00000 0.18454 -1.96595 -0.54733 1.86494 0.89148 -1.70043 -1.20527 1.47802 1.47802 -1.20527 -1.70043 0.89148 1.86494 -0.54733 -1.96595 0.18454 0.00000 ... ... ..... ....... ......... ........... ............. E13g 2.00000 -0.06159 -1.99621 0.18454 1.98484 -0.30678 -1.96595 0.42787 1.93959 -0.54733 -1.90588 0.66471 1.86494 -0.77957 -1.81693 0.89148 1.76202 -1.00000 -1.70043 1.10473 1.63239 -1.20527 -1.55816 1.30124 1.47802 -1.39227 0.00000 2.00000 -1.39227 1.47802 1.30124 -1.55816 -1.20527 1.63239 1.10473 -1.70043 -1.00000 1.76202 0.89148 -1.81693 -0.77957 1.86494 0.66471 -1.90588 -0.54733 1.93959 0.42787 -1.96595 -0.30678 1.98484 0.18454 -1.99621 -0.06159 0.00000 ... ... ..... ....... ......... ........... ............. E14g 2.00000 -0.30678 -1.90588 0.89148 1.63239 -1.39227 -1.20527 1.76202 0.66471 -1.96595 -0.06159 1.98484 -0.54733 -1.81693 1.10473 1.47802 -1.55816 -1.00000 1.86494 0.42787 -1.99621 0.18454 1.93959 -0.77957 -1.70043 1.30124 0.00000 2.00000 1.30124 -1.70043 -0.77957 1.93959 0.18454 -1.99621 0.42787 1.86494 -1.00000 -1.55816 1.47802 1.10473 -1.81693 -0.54733 1.98484 -0.06159 -1.96595 0.66471 1.76202 -1.20527 -1.39227 1.63239 0.89148 -1.90588 -0.30678 0.00000 ... ... ..... ....... ......... ........... ............. E15g 2.00000 -0.54733 -1.70043 1.47802 0.89148 -1.96595 0.18454 1.86494 -1.20527 -1.20527 1.86494 0.18454 -1.96595 0.89148 1.47802 -1.70043 -0.54733 2.00000 -0.54733 -1.70043 1.47802 0.89148 -1.96595 0.18454 1.86494 -1.20527 0.00000 2.00000 -1.20527 1.86494 0.18454 -1.96595 0.89148 1.47802 -1.70043 -0.54733 2.00000 -0.54733 -1.70043 1.47802 0.89148 -1.96595 0.18454 1.86494 -1.20527 -1.20527 1.86494 0.18454 -1.96595 0.89148 1.47802 -1.70043 -0.54733 0.00000 ... ... ..... ....... ......... ........... ............. E16g 2.00000 -0.77957 -1.39227 1.86494 -0.06159 -1.81693 1.47802 0.66471 -1.99621 0.89148 1.30124 -1.90588 0.18454 1.76202 -1.55816 -0.54733 1.98484 -1.00000 -1.20527 1.93959 -0.30678 -1.70043 1.63239 0.42787 -1.96595 1.10473 0.00000 2.00000 1.10473 -1.96595 0.42787 1.63239 -1.70043 -0.30678 1.93959 -1.20527 -1.00000 1.98484 -0.54733 -1.55816 1.76202 0.18454 -1.90588 1.30124 0.89148 -1.99621 0.66471 1.47802 -1.81693 -0.06159 1.86494 -1.39227 -0.77957 0.00000 ... ... ..... ....... ......... ........... ............. E17g 2.00000 -1.00000 -1.00000 2.00000 -1.00000 -1.00000 2.00000 -1.00000 -1.00000 2.00000 -1.00000 -1.00000 2.00000 -1.00000 -1.00000 2.00000 -1.00000 -1.00000 2.00000 -1.00000 -1.00000 2.00000 -1.00000 -1.00000 2.00000 -1.00000 0.00000 2.00000 -1.00000 2.00000 -1.00000 -1.00000 2.00000 -1.00000 -1.00000 2.00000 -1.00000 -1.00000 2.00000 -1.00000 -1.00000 2.00000 -1.00000 -1.00000 2.00000 -1.00000 -1.00000 2.00000 -1.00000 -1.00000 2.00000 -1.00000 -1.00000 0.00000 ... ... ..... ....... ......... ........... ............. E18g 2.00000 -1.20527 -0.54733 1.86494 -1.70043 0.18454 1.47802 -1.96595 0.89148 0.89148 -1.96595 1.47802 0.18454 -1.70043 1.86494 -0.54733 -1.20527 2.00000 -1.20527 -0.54733 1.86494 -1.70043 0.18454 1.47802 -1.96595 0.89148 0.00000 2.00000 0.89148 -1.96595 1.47802 0.18454 -1.70043 1.86494 -0.54733 -1.20527 2.00000 -1.20527 -0.54733 1.86494 -1.70043 0.18454 1.47802 -1.96595 0.89148 0.89148 -1.96595 1.47802 0.18454 -1.70043 1.86494 -0.54733 -1.20527 0.00000 ... ... ..... ....... ......... ........... ............. E19g 2.00000 -1.39227 -0.06159 1.47802 -1.99621 1.30124 0.18454 -1.55816 1.98484 -1.20527 -0.30678 1.63239 -1.96595 1.10473 0.42787 -1.70043 1.93959 -1.00000 -0.54733 1.76202 -1.90588 0.89148 0.66471 -1.81693 1.86494 -0.77957 0.00000 2.00000 -0.77957 1.86494 -1.81693 0.66471 0.89148 -1.90588 1.76202 -0.54733 -1.00000 1.93959 -1.70043 0.42787 1.10473 -1.96595 1.63239 -0.30678 -1.20527 1.98484 -1.55816 0.18454 1.30124 -1.99621 1.47802 -0.06159 -1.39227 0.00000 ... ... ..... ....... ......... ........... ............. E20g 2.00000 -1.55816 0.42787 0.89148 -1.81693 1.93959 -1.20527 -0.06159 1.30124 -1.96595 1.76202 -0.77957 -0.54733 1.63239 -1.99621 1.47802 -0.30678 -1.00000 1.86494 -1.90588 1.10473 0.18454 -1.39227 1.98484 -1.70043 0.66471 0.00000 2.00000 0.66471 -1.70043 1.98484 -1.39227 0.18454 1.10473 -1.90588 1.86494 -1.00000 -0.30678 1.47802 -1.99621 1.63239 -0.54733 -0.77957 1.76202 -1.96595 1.30124 -0.06159 -1.20527 1.93959 -1.81693 0.89148 0.42787 -1.55816 0.00000 ... ... ..... ....... ......... ........... ............. E21g 2.00000 -1.70043 0.89148 0.18454 -1.20527 1.86494 -1.96595 1.47802 -0.54733 -0.54733 1.47802 -1.96595 1.86494 -1.20527 0.18454 0.89148 -1.70043 2.00000 -1.70043 0.89148 0.18454 -1.20527 1.86494 -1.96595 1.47802 -0.54733 0.00000 2.00000 -0.54733 1.47802 -1.96595 1.86494 -1.20527 0.18454 0.89148 -1.70043 2.00000 -1.70043 0.89148 0.18454 -1.20527 1.86494 -1.96595 1.47802 -0.54733 -0.54733 1.47802 -1.96595 1.86494 -1.20527 0.18454 0.89148 -1.70043 0.00000 ... ... ..... ....... ......... ........... ............. E22g 2.00000 -1.81693 1.30124 -0.54733 -0.30678 1.10473 -1.70043 1.98484 -1.90588 1.47802 -0.77957 -0.06159 0.89148 -1.55816 1.93959 -1.96595 1.63239 -1.00000 0.18454 0.66471 -1.39227 1.86494 -1.99621 1.76202 -1.20527 0.42787 0.00000 2.00000 0.42787 -1.20527 1.76202 -1.99621 1.86494 -1.39227 0.66471 0.18454 -1.00000 1.63239 -1.96595 1.93959 -1.55816 0.89148 -0.06159 -0.77957 1.47802 -1.90588 1.98484 -1.70043 1.10473 -0.30678 -0.54733 1.30124 -1.81693 0.00000 ... ... ..... ....... ......... ........... ............. E23g 2.00000 -1.90588 1.63239 -1.20527 0.66471 -0.06159 -0.54733 1.10473 -1.55816 1.86494 -1.99621 1.93959 -1.70043 1.30124 -0.77957 0.18454 0.42787 -1.00000 1.47802 -1.81693 1.98484 -1.96595 1.76202 -1.39227 0.89148 -0.30678 0.00000 2.00000 -0.30678 0.89148 -1.39227 1.76202 -1.96595 1.98484 -1.81693 1.47802 -1.00000 0.42787 0.18454 -0.77957 1.30124 -1.70043 1.93959 -1.99621 1.86494 -1.55816 1.10473 -0.54733 -0.06159 0.66471 -1.20527 1.63239 -1.90588 0.00000 ... ... ..... ....... ......... ........... ............. E24g 2.00000 -1.96595 1.86494 -1.70043 1.47802 -1.20527 0.89148 -0.54733 0.18454 0.18454 -0.54733 0.89148 -1.20527 1.47802 -1.70043 1.86494 -1.96595 2.00000 -1.96595 1.86494 -1.70043 1.47802 -1.20527 0.89148 -0.54733 0.18454 0.00000 2.00000 0.18454 -0.54733 0.89148 -1.20527 1.47802 -1.70043 1.86494 -1.96595 2.00000 -1.96595 1.86494 -1.70043 1.47802 -1.20527 0.89148 -0.54733 0.18454 0.18454 -0.54733 0.89148 -1.20527 1.47802 -1.70043 1.86494 -1.96595 0.00000 ... ... ..... ....... ......... ........... ............. E25g 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 2.00000 -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 0.00000 ... ... ..... ....... ......... ........... ............. A1u 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 ... ... ..... ....... ......... ........... ............. A2u 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -1.00000 -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 ............. E1u 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 -2.00000 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 0.00000 ... TT. ..... ....TT. ......... ........TT. ............. E2u 2.00000 1.93959 1.76202 1.47802 1.10473 0.66471 0.18454 -0.30678 -0.77957 -1.20527 -1.55816 -1.81693 -1.96595 -1.99621 -1.90588 -1.70043 -1.39227 -1.00000 -0.54733 -0.06159 0.42787 0.89148 1.30124 1.63239 1.86494 1.98484 0.00000 -2.00000 -1.98484 -1.86494 -1.63239 -1.30124 -0.89148 -0.42787 0.06159 0.54733 1.00000 1.39227 1.70043 1.90588 1.99621 1.96595 1.81693 1.55816 1.20527 0.77957 0.30678 -0.18454 -0.66471 -1.10473 -1.47802 -1.76202 -1.93959 0.00000 ... ... ..... ..TT... ......... ......TT... ............. E3u 2.00000 1.86494 1.47802 0.89148 0.18454 -0.54733 -1.20527 -1.70043 -1.96595 -1.96595 -1.70043 -1.20527 -0.54733 0.18454 0.89148 1.47802 1.86494 2.00000 1.86494 1.47802 0.89148 0.18454 -0.54733 -1.20527 -1.70043 -1.96595 0.00000 -2.00000 1.96595 1.70043 1.20527 0.54733 -0.18454 -0.89148 -1.47802 -1.86494 -2.00000 -1.86494 -1.47802 -0.89148 -0.18454 0.54733 1.20527 1.70043 1.96595 1.96595 1.70043 1.20527 0.54733 -0.18454 -0.89148 -1.47802 -1.86494 0.00000 ... ... ..... TT..... ......... ....TT..... ............. E4u 2.00000 1.76202 1.10473 0.18454 -0.77957 -1.55816 -1.96595 -1.90588 -1.39227 -0.54733 0.42787 1.30124 1.86494 1.98484 1.63239 0.89148 -0.06159 -1.00000 -1.70043 -1.99621 -1.81693 -1.20527 -0.30678 0.66471 1.47802 1.93959 0.00000 -2.00000 -1.93959 -1.47802 -0.66471 0.30678 1.20527 1.81693 1.99621 1.70043 1.00000 0.06159 -0.89148 -1.63239 -1.98484 -1.86494 -1.30124 -0.42787 0.54733 1.39227 1.90588 1.96595 1.55816 0.77957 -0.18454 -1.10473 -1.76202 0.00000 ... ... ..... ....... ......... ..TT....... ............. E5u 2.00000 1.63239 0.66471 -0.54733 -1.55816 -1.99621 -1.70043 -0.77957 0.42787 1.47802 1.98484 1.76202 0.89148 -0.30678 -1.39227 -1.96595 -1.81693 -1.00000 0.18454 1.30124 1.93959 1.86494 1.10473 -0.06159 -1.20527 -1.90588 0.00000 -2.00000 1.90588 1.20527 0.06159 -1.10473 -1.86494 -1.93959 -1.30124 -0.18454 1.00000 1.81693 1.96595 1.39227 0.30678 -0.89148 -1.76202 -1.98484 -1.47802 -0.42787 0.77957 1.70043 1.99621 1.55816 0.54733 -0.66471 -1.63239 0.00000 ... ... ..... ....... ......... TT......... ............. E6u 2.00000 1.47802 0.18454 -1.20527 -1.96595 -1.70043 -0.54733 0.89148 1.86494 1.86494 0.89148 -0.54733 -1.70043 -1.96595 -1.20527 0.18454 1.47802 2.00000 1.47802 0.18454 -1.20527 -1.96595 -1.70043 -0.54733 0.89148 1.86494 0.00000 -2.00000 -1.86494 -0.89148 0.54733 1.70043 1.96595 1.20527 -0.18454 -1.47802 -2.00000 -1.47802 -0.18454 1.20527 1.96595 1.70043 0.54733 -0.89148 -1.86494 -1.86494 -0.89148 0.54733 1.70043 1.96595 1.20527 -0.18454 -1.47802 0.00000 ... ... ..... ....... ......... ........... ............. E7u 2.00000 1.30124 -0.30678 -1.70043 -1.90588 -0.77957 0.89148 1.93959 1.63239 0.18454 -1.39227 -1.99621 -1.20527 0.42787 1.76202 1.86494 0.66471 -1.00000 -1.96595 -1.55816 -0.06159 1.47802 1.98484 1.10473 -0.54733 -1.81693 0.00000 -2.00000 1.81693 0.54733 -1.10473 -1.98484 -1.47802 0.06159 1.55816 1.96595 1.00000 -0.66471 -1.86494 -1.76202 -0.42787 1.20527 1.99621 1.39227 -0.18454 -1.63239 -1.93959 -0.89148 0.77957 1.90588 1.70043 0.30678 -1.30124 0.00000 ... ... ..... ....... ......... ........... ............. E8u 2.00000 1.10473 -0.77957 -1.96595 -1.39227 0.42787 1.86494 1.63239 -0.06159 -1.70043 -1.81693 -0.30678 1.47802 1.93959 0.66471 -1.20527 -1.99621 -1.00000 0.89148 1.98484 1.30124 -0.54733 -1.90588 -1.55816 0.18454 1.76202 0.00000 -2.00000 -1.76202 -0.18454 1.55816 1.90588 0.54733 -1.30124 -1.98484 -0.89148 1.00000 1.99621 1.20527 -0.66471 -1.93959 -1.47802 0.30678 1.81693 1.70043 0.06159 -1.63239 -1.86494 -0.42787 1.39227 1.96595 0.77957 -1.10473 0.00000 ... ... ..... ....... ......... ........... ............. 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E10u 2.00000 0.66471 -1.55816 -1.70043 0.42787 1.98484 0.89148 -1.39227 -1.81693 0.18454 1.93959 1.10473 -1.20527 -1.90588 -0.06159 1.86494 1.30124 -1.00000 -1.96595 -0.30678 1.76202 1.47802 -0.77957 -1.99621 -0.54733 1.63239 0.00000 -2.00000 -1.63239 0.54733 1.99621 0.77957 -1.47802 -1.76202 0.30678 1.96595 1.00000 -1.30124 -1.86494 0.06159 1.90588 1.20527 -1.10473 -1.93959 -0.18454 1.81693 1.39227 -0.89148 -1.98484 -0.42787 1.70043 1.55816 -0.66471 0.00000 ... ... ..... ....... ......... ........... ............. 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E13u 2.00000 -0.06159 -1.99621 0.18454 1.98484 -0.30678 -1.96595 0.42787 1.93959 -0.54733 -1.90588 0.66471 1.86494 -0.77957 -1.81693 0.89148 1.76202 -1.00000 -1.70043 1.10473 1.63239 -1.20527 -1.55816 1.30124 1.47802 -1.39227 0.00000 -2.00000 1.39227 -1.47802 -1.30124 1.55816 1.20527 -1.63239 -1.10473 1.70043 1.00000 -1.76202 -0.89148 1.81693 0.77957 -1.86494 -0.66471 1.90588 0.54733 -1.93959 -0.42787 1.96595 0.30678 -1.98484 -0.18454 1.99621 0.06159 0.00000 ... ... ..... ....... ......... ........... ............. E14u 2.00000 -0.30678 -1.90588 0.89148 1.63239 -1.39227 -1.20527 1.76202 0.66471 -1.96595 -0.06159 1.98484 -0.54733 -1.81693 1.10473 1.47802 -1.55816 -1.00000 1.86494 0.42787 -1.99621 0.18454 1.93959 -0.77957 -1.70043 1.30124 0.00000 -2.00000 -1.30124 1.70043 0.77957 -1.93959 -0.18454 1.99621 -0.42787 -1.86494 1.00000 1.55816 -1.47802 -1.10473 1.81693 0.54733 -1.98484 0.06159 1.96595 -0.66471 -1.76202 1.20527 1.39227 -1.63239 -0.89148 1.90588 0.30678 0.00000 ... ... ..... ....... ......... ........... ............. 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E24u 2.00000 -1.96595 1.86494 -1.70043 1.47802 -1.20527 0.89148 -0.54733 0.18454 0.18454 -0.54733 0.89148 -1.20527 1.47802 -1.70043 1.86494 -1.96595 2.00000 -1.96595 1.86494 -1.70043 1.47802 -1.20527 0.89148 -0.54733 0.18454 0.00000 -2.00000 -0.18454 0.54733 -0.89148 1.20527 -1.47802 1.70043 -1.86494 1.96595 -2.00000 1.96595 -1.86494 1.70043 -1.47802 1.20527 -0.89148 0.54733 -0.18454 -0.18454 0.54733 -0.89148 1.20527 -1.47802 1.70043 -1.86494 1.96595 0.00000 ... ... ..... ....... ......... ........... ............. E25u 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 -2.00000 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 0.00000 ... ... ..... ....... ......... ........... ............. Irrational character values: 1.996206657474 = 2*cos(2*π/102) = 2*cos(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 1.984841019344 = 2*cos(4*π/102) = 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 1.965946199368 = 2*cos(6*π/102) = 2*cos(π/17) = (1−√17+√34−2*√17+2*√17+3*√17+√34−2*√17+2*√34+2*√17)/8 1.939593872070 = 2*cos(8*π/102) = 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 1.905884000854 = 2*cos(10*π/102) = 2*cos(5*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 1.864944458809 = 2*cos(12*π/102) = 2*cos(2*π/17) = (−1+√17+√34−2*√17+2*√17+3*√17−√34−2*√17−2*√34+2*√17)/8 1.816930543639 = 2*cos(14*π/102) = 2*cos(7*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 1.762024388572 = 2*cos(16*π/102) = 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 1.700434271459 = 2*cos(18*π/102) = 2*cos(3*π/17) = (1+√17+√34+2*√17+2*√17−3*√17+√34+2*√17−2*√34−2*√17)/8 1.632393824712 = 2*cos(20*π/102) = 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 1.558161149051 = 2*cos(22*π/102) = 2*cos(11*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 1.478017834441 = 2*cos(24*π/102) = 2*cos(4*π/17) = (−1+√17−√34−2*√17+2*√17+3*√17+√34−2*√17+2*√34+2*√17)/8 1.392267891926 = 2*cos(26*π/102) = 2*cos(13*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 1.301236600408 = 2*cos(28*π/102) = 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 1.205269272759 = 2*cos(30*π/102) = 2*cos(5*π/17) = (1+√17+√34+2*√17−2*√17−3*√17+√34+2*√17−2*√34−2*√17)/8 1.104729945921 = 2*cos(32*π/102) = 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 0.891476711553 = 2*cos(36*π/102) = 2*cos(6*π/17) = (−1−√17+√34+2*√17+2*√17−3*√17−√34+2*√17+2*√34−2*√17)/8 0.779571746585 = 2*cos(38*π/102) = 2*cos(19*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 0.664709598959 = 2*cos(40*π/102) = 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 0.547325980144 = 2*cos(42*π/102) = 2*cos(7*π/17) = (1+√17−√34+2*√17+2*√17−3*√17−√34+2*√17+2*√34−2*√17)/8 0.427866166413 = 2*cos(44*π/102) = 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 0.306783309757 = 2*cos(46*π/102) = 2*cos(23*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 0.184536718927 = 2*cos(48*π/102) = 2*cos(8*π/17) = (−1+√17+√34−2*√17−2*√17+3*√17−√34−2*√17−2*√34+2*√17)/8 0.061590117112 = 2*cos(50*π/102) = 2*cos(25*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 Symmetry of Rotations and Cartesian products A1g d+g+i+k+m z², z⁴, z⁶ A2g R R_z E1g R+d+g+i+k+m {R_x, R_y}, {xz, yz}, {xz³, yz³}, {xz⁵, yz⁵} E2g d+g+i+k+m {x²−y², xy}, {z²(x²−y²), xyz²}, {z⁴(x²−y²), xyz⁴} E3g g+i+k+m {xz(x²−3y²), yz(3x²−y²)}, {xz³(x²−3y²), yz³(3x²−y²)} E4g g+i+k+m {(x²−y²)²−4x²y², xy(x²−y²)}, {z²((x²−y²)²−4x²y²), xyz²(x²−y²)} E5g 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)} E6g i+k+m {x²(x²−3y²)²−y²(3x²−y²)², xy(x²−3y²)(3x²−y²)} E7g k+m E8g k+m E9g m E10g m A2u p+f+h+j+l z, z³, z⁵ E1u p+f+h+j+l {x, y}, {xz², yz²}, {xz⁴, yz⁴} E2u f+h+j+l {z(x²−y²), xyz}, {z³(x²−y²), xyz³} E3u f+h+j+l {x(x²−3y²), y(3x²−y²)}, {xz²(x²−3y²), yz²(3x²−y²)} E4u h+j+l {z((x²−y²)²−4x²y²), xyz(x²−y²)} E5u 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)} E6u j+l E7u j+l E8u l E9u l Notes: α The order of the D_51d point group is 204, and the order of the principal axis (S₁₀₂) is 102. The group has 54 irreducible representations. β The D_51d point group is isomorphic to D_51h, C_102v and D₁₀₂. γ The D_51d point group is generated by two symmetry elements, S₁₀₂ 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). The canonical choice, however, is to use redundant generators: C₅₁, C₂^′ and i. δ The group contains one set of C₂^′ symmetry axes perpendicular to the principal (z) axis. The x axis (but not the y axis) is a member of that set. Reversely, the single set of symmetry planes denoted σ_d contains the yz plane but not the xz plane. ε The lowest nonvanishing multipole moment in D_51d is 4 (quadrupole 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 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. 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 51-gon can be derived from the regular 17-gon by overlaying it with a equilateral triangle.

This Character Table for the D_51d point group was created by Gernot Katzer.

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

	D_49d
	D_50d
C₅₁ C_51v C_51h D₅₁ D_51h	D_51d
	D_52d
	D_53d

Character table for the D51d point group

Character table for the D_51d point group