Character table for the C109 point group

C109    E         2 C109    2 C109^2  2 C109^3  2 C109^4  2 C109^5  2 C109^6  2 C109^7  2 C109^8  2 C109^9  2 C109^10 2 C109^11 2 C109^12 2 C109^13 2 C109^14 2 C109^15 2 C109^16 2 C109^17 2 C109^18 2 C109^19 2 C109^20 2 C109^21 2 C109^22 2 C109^23 2 C109^24 2 C109^25 2 C109^26 2 C109^27 2 C109^28 2 C109^29 2 C109^30 2 C109^31 2 C109^32 2 C109^33 2 C109^34 2 C109^35 2 C109^36 2 C109^37 2 C109^38 2 C109^39 2 C109^40 2 C109^41 2 C109^42 2 C109^43 2 C109^44 2 C109^45 2 C109^46 2 C109^47 2 C109^48 2 C109^49 2 C109^50 2 C109^51 2 C109^52 2 C109^53 2 C109^54    <R> <p> <—d—> <——f——> <———g———> <————h————> <—————i—————> 
A        1.00000   1.00000   1.00000   1.00000   1.00000   1.00000   1.00000   1.00000   1.00000   1.00000   1.00000   1.00000   1.00000   1.00000   1.00000   1.00000   1.00000   1.00000   1.00000   1.00000   1.00000   1.00000   1.00000   1.00000   1.00000   1.00000   1.00000   1.00000   1.00000   1.00000   1.00000   1.00000   1.00000   1.00000   1.00000   1.00000   1.00000   1.00000   1.00000   1.00000   1.00000   1.00000   1.00000   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 ............T
E1  *    2.00000   1.99668   1.98672   1.97017   1.94707   1.91750   1.88157   1.83938   1.79108   1.73683   1.67682   1.61123   1.54029   1.46424   1.38332   1.29780   1.20797   1.11413   1.01659   0.91568   0.81172   0.70506   0.59606   0.48509   0.37250   0.25867   0.14399   0.02882  -0.08644  -0.20141  -0.31572  -0.42897  -0.54080  -0.65083  -0.75870  -0.86406  -0.96654  -1.06581  -1.16154  -1.25341  -1.34111  -1.42437  -1.50289  -1.57642  -1.64471  -1.70754  -1.76469  -1.81598  -1.86125  -1.90032  -1.93309  -1.95943  -1.97927  -1.99253  -1.99917     TT. TT. ..TT. ....TT. ......TT. ........TT. ..........TT.
E2  *    2.00000   1.98672   1.94707   1.88157   1.79108   1.67682   1.54029   1.38332   1.20797   1.01659   0.81172   0.59606   0.37250   0.14399  -0.08644  -0.31572  -0.54080  -0.75870  -0.96654  -1.16154  -1.34111  -1.50289  -1.64471  -1.76469  -1.86125  -1.93309  -1.97927  -1.99917  -1.99253  -1.95943  -1.90032  -1.81598  -1.70754  -1.57642  -1.42437  -1.25341  -1.06581  -0.86406  -0.65083  -0.42897  -0.20141   0.02882   0.25867   0.48509   0.70506   0.91568   1.11413   1.29780   1.46424   1.61123   1.73683   1.83938   1.91750   1.97017   1.99668     ... ... TT... ..TT... ....TT... ......TT... ........TT...
E3  *    2.00000   1.97017   1.88157   1.73683   1.54029   1.29780   1.01659   0.70506   0.37250   0.02882  -0.31572  -0.65083  -0.96654  -1.25341  -1.50289  -1.70754  -1.86125  -1.95943  -1.99917  -1.97927  -1.90032  -1.76469  -1.57642  -1.34111  -1.06581  -0.75870  -0.42897  -0.08644   0.25867   0.59606   0.91568   1.20797   1.46424   1.67682   1.83938   1.94707   1.99668   1.98672   1.91750   1.79108   1.61123   1.38332   1.11413   0.81172   0.48509   0.14399  -0.20141  -0.54080  -0.86406  -1.16154  -1.42437  -1.64471  -1.81598  -1.93309  -1.99253     ... ... ..... TT..... ..TT..... ....TT..... ......TT.....
E4  *    2.00000   1.94707   1.79108   1.54029   1.20797   0.81172   0.37250  -0.08644  -0.54080  -0.96654  -1.34111  -1.64471  -1.86125  -1.97927  -1.99253  -1.90032  -1.70754  -1.42437  -1.06581  -0.65083  -0.20141   0.25867   0.70506   1.11413   1.46424   1.73683   1.91750   1.99668   1.97017   1.83938   1.61123   1.29780   0.91568   0.48509   0.02882  -0.42897  -0.86406  -1.25341  -1.57642  -1.81598  -1.95943  -1.99917  -1.93309  -1.76469  -1.50289  -1.16154  -0.75870  -0.31572   0.14399   0.59606   1.01659   1.38332   1.67682   1.88157   1.98672     ... ... ..... ....... TT....... ..TT....... ....TT.......
E5  *    2.00000   1.91750   1.67682   1.29780   0.81172   0.25867  -0.31572  -0.86406  -1.34111  -1.70754  -1.93309  -1.99917  -1.90032  -1.64471  -1.25341  -0.75870  -0.20141   0.37250   0.91568   1.38332   1.73683   1.94707   1.99668   1.88157   1.61123   1.20797   0.70506   0.14399  -0.42897  -0.96654  -1.42437  -1.76469  -1.95943  -1.99253  -1.86125  -1.57642  -1.16154  -0.65083  -0.08644   0.48509   1.01659   1.46424   1.79108   1.97017   1.98672   1.83938   1.54029   1.11413   0.59606   0.02882  -0.54080  -1.06581  -1.50289  -1.81598  -1.97927     ... ... ..... ....... ......... TT......... ..TT.........
E6  *    2.00000   1.88157   1.54029   1.01659   0.37250  -0.31572  -0.96654  -1.50289  -1.86125  -1.99917  -1.90032  -1.57642  -1.06581  -0.42897   0.25867   0.91568   1.46424   1.83938   1.99668   1.91750   1.61123   1.11413   0.48509  -0.20141  -0.86406  -1.42437  -1.81598  -1.99253  -1.93309  -1.64471  -1.16154  -0.54080   0.14399   0.81172   1.38332   1.79108   1.98672   1.94707   1.67682   1.20797   0.59606  -0.08644  -0.75870  -1.34111  -1.76469  -1.97927  -1.95943  -1.70754  -1.25341  -0.65083   0.02882   0.70506   1.29780   1.73683   1.97017     ... ... ..... ....... ......... ........... TT...........
E7  *    2.00000   1.83938   1.38332   0.70506  -0.08644  -0.86406  -1.50289  -1.90032  -1.99253  -1.76469  -1.25341  -0.54080   0.25867   1.01659   1.61123   1.94707   1.97017   1.67682   1.11413   0.37250  -0.42897  -1.16154  -1.70754  -1.97927  -1.93309  -1.57642  -0.96654  -0.20141   0.59606   1.29780   1.79108   1.99668   1.88157   1.46424   0.81172   0.02882  -0.75870  -1.42437  -1.86125  -1.99917  -1.81598  -1.34111  -0.65083   0.14399   0.91568   1.54029   1.91750   1.98672   1.73683   1.20797   0.48509  -0.31572  -1.06581  -1.64471  -1.95943     ... ... ..... ....... ......... ........... .............
E8  *    2.00000   1.79108   1.20797   0.37250  -0.54080  -1.34111  -1.86125  -1.99253  -1.70754  -1.06581  -0.20141   0.70506   1.46424   1.91750   1.97017   1.61123   0.91568   0.02882  -0.86406  -1.57642  -1.95943  -1.93309  -1.50289  -0.75870   0.14399   1.01659   1.67682   1.98672   1.88157   1.38332   0.59606  -0.31572  -1.16154  -1.76469  -1.99917  -1.81598  -1.25341  -0.42897   0.48509   1.29780   1.83938   1.99668   1.73683   1.11413   0.25867  -0.65083  -1.42437  -1.90032  -1.97927  -1.64471  -0.96654  -0.08644   0.81172   1.54029   1.94707     ... ... ..... ....... ......... ........... .............
E9  *    2.00000   1.73683   1.01659   0.02882  -0.96654  -1.70754  -1.99917  -1.76469  -1.06581  -0.08644   0.91568   1.67682   1.99668   1.79108   1.11413   0.14399  -0.86406  -1.64471  -1.99253  -1.81598  -1.16154  -0.20141   0.81172   1.61123   1.98672   1.83938   1.20797   0.25867  -0.75870  -1.57642  -1.97927  -1.86125  -1.25341  -0.31572   0.70506   1.54029   1.97017   1.88157   1.29780   0.37250  -0.65083  -1.50289  -1.95943  -1.90032  -1.34111  -0.42897   0.59606   1.46424   1.94707   1.91750   1.38332   0.48509  -0.54080  -1.42437  -1.93309     ... ... ..... ....... ......... ........... .............
E10 *    2.00000   1.67682   0.81172  -0.31572  -1.34111  -1.93309  -1.90032  -1.25341  -0.20141   0.91568   1.73683   1.99668   1.61123   0.70506  -0.42897  -1.42437  -1.95943  -1.86125  -1.16154  -0.08644   1.01659   1.79108   1.98672   1.54029   0.59606  -0.54080  -1.50289  -1.97927  -1.81598  -1.06581   0.02882   1.11413   1.83938   1.97017   1.46424   0.48509  -0.65083  -1.57642  -1.99253  -1.76469  -0.96654   0.14399   1.20797   1.88157   1.94707   1.38332   0.37250  -0.75870  -1.64471  -1.99917  -1.70754  -0.86406   0.25867   1.29780   1.91750     ... ... ..... ....... ......... ........... .............
E11 *    2.00000   1.61123   0.59606  -0.65083  -1.64471  -1.99917  -1.57642  -0.54080   0.70506   1.67682   1.99668   1.54029   0.48509  -0.75870  -1.70754  -1.99253  -1.50289  -0.42897   0.81172   1.73683   1.98672   1.46424   0.37250  -0.86406  -1.76469  -1.97927  -1.42437  -0.31572   0.91568   1.79108   1.97017   1.38332   0.25867  -0.96654  -1.81598  -1.95943  -1.34111  -0.20141   1.01659   1.83938   1.94707   1.29780   0.14399  -1.06581  -1.86125  -1.93309  -1.25341  -0.08644   1.11413   1.88157   1.91750   1.20797   0.02882  -1.16154  -1.90032     ... ... ..... ....... ......... ........... .............
E12 *    2.00000   1.54029   0.37250  -0.96654  -1.86125  -1.90032  -1.06581   0.25867   1.46424   1.99668   1.61123   0.48509  -0.86406  -1.81598  -1.93309  -1.16154   0.14399   1.38332   1.98672   1.67682   0.59606  -0.75870  -1.76469  -1.95943  -1.25341   0.02882   1.29780   1.97017   1.73683   0.70506  -0.65083  -1.70754  -1.97927  -1.34111  -0.08644   1.20797   1.94707   1.79108   0.81172  -0.54080  -1.64471  -1.99253  -1.42437  -0.20141   1.11413   1.91750   1.83938   0.91568  -0.42897  -1.57642  -1.99917  -1.50289  -0.31572   1.01659   1.88157     ... ... ..... ....... ......... ........... .............
E13 *    2.00000   1.46424   0.14399  -1.25341  -1.97927  -1.64471  -0.42897   1.01659   1.91750   1.79108   0.70506  -0.75870  -1.81598  -1.90032  -0.96654   0.48509   1.67682   1.97017   1.20797  -0.20141  -1.50289  -1.99917  -1.42437  -0.08644   1.29780   1.98672   1.61123   0.37250  -1.06581  -1.93309  -1.76469  -0.65083   0.81172   1.83938   1.88157   0.91568  -0.54080  -1.70754  -1.95943  -1.16154   0.25867   1.54029   1.99668   1.38332   0.02882  -1.34111  -1.99253  -1.57642  -0.31572   1.11413   1.94707   1.73683   0.59606  -0.86406  -1.86125     ... ... ..... ....... ......... ........... .............
E14 *    2.00000   1.38332  -0.08644  -1.50289  -1.99253  -1.25341   0.25867   1.61123   1.97017   1.11413  -0.42897  -1.70754  -1.93309  -0.96654   0.59606   1.79108   1.88157   0.81172  -0.75870  -1.86125  -1.81598  -0.65083   0.91568   1.91750   1.73683   0.48509  -1.06581  -1.95943  -1.64471  -0.31572   1.20797   1.98672   1.54029   0.14399  -1.34111  -1.99917  -1.42437   0.02882   1.46424   1.99668   1.29780  -0.20141  -1.57642  -1.97927  -1.16154   0.37250   1.67682   1.94707   1.01659  -0.54080  -1.76469  -1.90032  -0.86406   0.70506   1.83938     ... ... ..... ....... ......... ........... .............
E15 *    2.00000   1.29780  -0.31572  -1.70754  -1.90032  -0.75870   0.91568   1.94707   1.61123   0.14399  -1.42437  -1.99253  -1.16154   0.48509   1.79108   1.83938   0.59606  -1.06581  -1.97927  -1.50289   0.02882   1.54029   1.97017   1.01659  -0.65083  -1.86125  -1.76469  -0.42897   1.20797   1.99668   1.38332  -0.20141  -1.64471  -1.93309  -0.86406   0.81172   1.91750   1.67682   0.25867  -1.34111  -1.99917  -1.25341   0.37250   1.73683   1.88157   0.70506  -0.96654  -1.95943  -1.57642  -0.08644   1.46424   1.98672   1.11413  -0.54080  -1.81598     ... ... ..... ....... ......... ........... .............
E16 *    2.00000   1.20797  -0.54080  -1.86125  -1.70754  -0.20141   1.46424   1.97017   0.91568  -0.86406  -1.95943  -1.50289   0.14399   1.67682   1.88157   0.59606  -1.16154  -1.99917  -1.25341   0.48509   1.83938   1.73683   0.25867  -1.42437  -1.97927  -0.96654   0.81172   1.94707   1.54029  -0.08644  -1.64471  -1.90032  -0.65083   1.11413   1.99668   1.29780  -0.42897  -1.81598  -1.76469  -0.31572   1.38332   1.98672   1.01659  -0.75870  -1.93309  -1.57642   0.02882   1.61123   1.91750   0.70506  -1.06581  -1.99253  -1.34111   0.37250   1.79108     ... ... ..... ....... ......... ........... .............
E17 *    2.00000   1.11413  -0.75870  -1.95943  -1.42437   0.37250   1.83938   1.67682   0.02882  -1.64471  -1.86125  -0.42897   1.38332   1.97017   0.81172  -1.06581  -1.99917  -1.16154   0.70506   1.94707   1.46424  -0.31572  -1.81598  -1.70754  -0.08644   1.61123   1.88157   0.48509  -1.34111  -1.97927  -0.86406   1.01659   1.99668   1.20797  -0.65083  -1.93309  -1.50289   0.25867   1.79108   1.73683   0.14399  -1.57642  -1.90032  -0.54080   1.29780   1.98672   0.91568  -0.96654  -1.99253  -1.25341   0.59606   1.91750   1.54029  -0.20141  -1.76469     ... ... ..... ....... ......... ........... .............
E18 *    2.00000   1.01659  -0.96654  -1.99917  -1.06581   0.91568   1.99668   1.11413  -0.86406  -1.99253  -1.16154   0.81172   1.98672   1.20797  -0.75870  -1.97927  -1.25341   0.70506   1.97017   1.29780  -0.65083  -1.95943  -1.34111   0.59606   1.94707   1.38332  -0.54080  -1.93309  -1.42437   0.48509   1.91750   1.46424  -0.42897  -1.90032  -1.50289   0.37250   1.88157   1.54029  -0.31572  -1.86125  -1.57642   0.25867   1.83938   1.61123  -0.20141  -1.81598  -1.64471   0.14399   1.79108   1.67682  -0.08644  -1.76469  -1.70754   0.02882   1.73683     ... ... ..... ....... ......... ........... .............
E19 *    2.00000   0.91568  -1.16154  -1.97927  -0.65083   1.38332   1.91750   0.37250  -1.57642  -1.81598  -0.08644   1.73683   1.67682  -0.20141  -1.86125  -1.50289   0.48509   1.94707   1.29780  -0.75870  -1.99253  -1.06581   1.01659   1.99668   0.81172  -1.25341  -1.95943  -0.54080   1.46424   1.88157   0.25867  -1.64471  -1.76469   0.02882   1.79108   1.61123  -0.31572  -1.90032  -1.42437   0.59606   1.97017   1.20797  -0.86406  -1.99917  -0.96654   1.11413   1.98672   0.70506  -1.34111  -1.93309  -0.42897   1.54029   1.83938   0.14399  -1.70754     ... ... ..... ....... ......... ........... .............
E20 *    2.00000   0.81172  -1.34111  -1.90032  -0.20141   1.73683   1.61123  -0.42897  -1.95943  -1.16154   1.01659   1.98672   0.59606  -1.50289  -1.81598   0.02882   1.83938   1.46424  -0.65083  -1.99253  -0.96654   1.20797   1.94707   0.37250  -1.64471  -1.70754   0.25867   1.91750   1.29780  -0.86406  -1.99917  -0.75870   1.38332   1.88157   0.14399  -1.76469  -1.57642   0.48509   1.97017   1.11413  -1.06581  -1.97927  -0.54080   1.54029   1.79108  -0.08644  -1.86125  -1.42437   0.70506   1.99668   0.91568  -1.25341  -1.93309  -0.31572   1.67682     ... ... ..... ....... ......... ........... .............
E21 *    2.00000   0.70506  -1.50289  -1.76469   0.25867   1.94707   1.11413  -1.16154  -1.93309  -0.20141   1.79108   1.46424  -0.75870  -1.99917  -0.65083   1.54029   1.73683  -0.31572  -1.95943  -1.06581   1.20797   1.91750   0.14399  -1.81598  -1.42437   0.81172   1.99668   0.59606  -1.57642  -1.70754   0.37250   1.97017   1.01659  -1.25341  -1.90032  -0.08644   1.83938   1.38332  -0.86406  -1.99253  -0.54080   1.61123   1.67682  -0.42897  -1.97927  -0.96654   1.29780   1.88157   0.02882  -1.86125  -1.34111   0.91568   1.98672   0.48509  -1.64471     ... ... ..... ....... ......... ........... .............
E22 *    2.00000   0.59606  -1.64471  -1.57642   0.70506   1.99668   0.48509  -1.70754  -1.50289   0.81172   1.98672   0.37250  -1.76469  -1.42437   0.91568   1.97017   0.25867  -1.81598  -1.34111   1.01659   1.94707   0.14399  -1.86125  -1.25341   1.11413   1.91750   0.02882  -1.90032  -1.16154   1.20797   1.88157  -0.08644  -1.93309  -1.06581   1.29780   1.83938  -0.20141  -1.95943  -0.96654   1.38332   1.79108  -0.31572  -1.97927  -0.86406   1.46424   1.73683  -0.42897  -1.99253  -0.75870   1.54029   1.67682  -0.54080  -1.99917  -0.65083   1.61123     ... ... ..... ....... ......... ........... .............
E23 *    2.00000   0.48509  -1.76469  -1.34111   1.11413   1.88157  -0.20141  -1.97927  -0.75870   1.61123   1.54029  -0.86406  -1.95943  -0.08644   1.91750   1.01659  -1.42437  -1.70754   0.59606   1.99668   0.37250  -1.81598  -1.25341   1.20797   1.83938  -0.31572  -1.99253  -0.65083   1.67682   1.46424  -0.96654  -1.93309   0.02882   1.94707   0.91568  -1.50289  -1.64471   0.70506   1.98672   0.25867  -1.86125  -1.16154   1.29780   1.79108  -0.42897  -1.99917  -0.54080   1.73683   1.38332  -1.06581  -1.90032   0.14399   1.97017   0.81172  -1.57642     ... ... ..... ....... ......... ........... .............
E24 *    2.00000   0.37250  -1.86125  -1.06581   1.46424   1.61123  -0.86406  -1.93309   0.14399   1.98672   0.59606  -1.76469  -1.25341   1.29780   1.73683  -0.65083  -1.97927  -0.08644   1.94707   0.81172  -1.64471  -1.42437   1.11413   1.83938  -0.42897  -1.99917  -0.31572   1.88157   1.01659  -1.50289  -1.57642   0.91568   1.91750  -0.20141  -1.99253  -0.54080   1.79108   1.20797  -1.34111  -1.70754   0.70506   1.97017   0.02882  -1.95943  -0.75870   1.67682   1.38332  -1.16154  -1.81598   0.48509   1.99668   0.25867  -1.90032  -0.96654   1.54029     ... ... ..... ....... ......... ........... .............
E25 *    2.00000   0.25867  -1.93309  -0.75870   1.73683   1.20797  -1.42437  -1.57642   1.01659   1.83938  -0.54080  -1.97927   0.02882   1.98672   0.48509  -1.86125  -0.96654   1.61123   1.38332  -1.25341  -1.70754   0.81172   1.91750  -0.31572  -1.99917  -0.20141   1.94707   0.70506  -1.76469  -1.16154   1.46424   1.54029  -1.06581  -1.81598   0.59606   1.97017  -0.08644  -1.99253  -0.42897   1.88157   0.91568  -1.64471  -1.34111   1.29780   1.67682  -0.86406  -1.90032   0.37250   1.99668   0.14399  -1.95943  -0.65083   1.79108   1.11413  -1.50289     ... ... ..... ....... ......... ........... .............
E26 *    2.00000   0.14399  -1.97927  -0.42897   1.91750   0.70506  -1.81598  -0.96654   1.67682   1.20797  -1.50289  -1.42437   1.29780   1.61123  -1.06581  -1.76469   0.81172   1.88157  -0.54080  -1.95943   0.25867   1.99668   0.02882  -1.99253  -0.31572   1.94707   0.59606  -1.86125  -0.86406   1.73683   1.11413  -1.57642  -1.34111   1.38332   1.54029  -1.16154  -1.70754   0.91568   1.83938  -0.65083  -1.93309   0.37250   1.98672  -0.08644  -1.99917  -0.20141   1.97017   0.48509  -1.90032  -0.75870   1.79108   1.01659  -1.64471  -1.25341   1.46424     ... ... ..... ....... ......... ........... .............
E27 *    2.00000   0.02882  -1.99917  -0.08644   1.99668   0.14399  -1.99253  -0.20141   1.98672   0.25867  -1.97927  -0.31572   1.97017   0.37250  -1.95943  -0.42897   1.94707   0.48509  -1.93309  -0.54080   1.91750   0.59606  -1.90032  -0.65083   1.88157   0.70506  -1.86125  -0.75870   1.83938   0.81172  -1.81598  -0.86406   1.79108   0.91568  -1.76469  -0.96654   1.73683   1.01659  -1.70754  -1.06581   1.67682   1.11413  -1.64471  -1.16154   1.61123   1.20797  -1.57642  -1.25341   1.54029   1.29780  -1.50289  -1.34111   1.46424   1.38332  -1.42437     ... ... ..... ....... ......... ........... .............
E28 *    2.00000  -0.08644  -1.99253   0.25867   1.97017  -0.42897  -1.93309   0.59606   1.88157  -0.75870  -1.81598   0.91568   1.73683  -1.06581  -1.64471   1.20797   1.54029  -1.34111  -1.42437   1.46424   1.29780  -1.57642  -1.16154   1.67682   1.01659  -1.76469  -0.86406   1.83938   0.70506  -1.90032  -0.54080   1.94707   0.37250  -1.97927  -0.20141   1.99668   0.02882  -1.99917   0.14399   1.98672  -0.31572  -1.95943   0.48509   1.91750  -0.65083  -1.86125   0.81172   1.79108  -0.96654  -1.70754   1.11413   1.61123  -1.25341  -1.50289   1.38332     ... ... ..... ....... ......... ........... .............
E29 *    2.00000  -0.20141  -1.95943   0.59606   1.83938  -0.96654  -1.64471   1.29780   1.38332  -1.57642  -1.06581   1.79108   0.70506  -1.93309  -0.31572   1.99668  -0.08644  -1.97927   0.48509   1.88157  -0.86406  -1.70754   1.20797   1.46424  -1.50289  -1.16154   1.73683   0.81172  -1.90032  -0.42897   1.98672   0.02882  -1.99253   0.37250   1.91750  -0.75870  -1.76469   1.11413   1.54029  -1.42437  -1.25341   1.67682   0.91568  -1.86125  -0.54080   1.97017   0.14399  -1.99917   0.25867   1.94707  -0.65083  -1.81598   1.01659   1.61123  -1.34111     ... ... ..... ....... ......... ........... .............
E30 *    2.00000  -0.31572  -1.90032   0.91568   1.61123  -1.42437  -1.16154   1.79108   0.59606  -1.97927   0.02882   1.97017  -0.65083  -1.76469   1.20797   1.38332  -1.64471  -0.86406   1.91750   0.25867  -1.99917   0.37250   1.88157  -0.96654  -1.57642   1.46424   1.11413  -1.81598  -0.54080   1.98672  -0.08644  -1.95943   0.70506   1.73683  -1.25341  -1.34111   1.67682   0.81172  -1.93309  -0.20141   1.99668  -0.42897  -1.86125   1.01659   1.54029  -1.50289  -1.06581   1.83938   0.48509  -1.99253   0.14399   1.94707  -0.75870  -1.70754   1.29780     ... ... ..... ....... ......... ........... .............
E31 *    2.00000  -0.42897  -1.81598   1.20797   1.29780  -1.76469  -0.54080   1.99668  -0.31572  -1.86125   1.11413   1.38332  -1.70754  -0.65083   1.98672  -0.20141  -1.90032   1.01659   1.46424  -1.64471  -0.75870   1.97017  -0.08644  -1.93309   0.91568   1.54029  -1.57642  -0.86406   1.94707   0.02882  -1.95943   0.81172   1.61123  -1.50289  -0.96654   1.91750   0.14399  -1.97927   0.70506   1.67682  -1.42437  -1.06581   1.88157   0.25867  -1.99253   0.59606   1.73683  -1.34111  -1.16154   1.83938   0.37250  -1.99917   0.48509   1.79108  -1.25341     ... ... ..... ....... ......... ........... .............
E32 *    2.00000  -0.54080  -1.70754   1.46424   0.91568  -1.95943   0.14399   1.88157  -1.16154  -1.25341   1.83938   0.25867  -1.97927   0.81172   1.54029  -1.64471  -0.65083   1.99668  -0.42897  -1.76469   1.38332   1.01659  -1.93309   0.02882   1.91750  -1.06581  -1.34111   1.79108   0.37250  -1.99253   0.70506   1.61123  -1.57642  -0.75870   1.98672  -0.31572  -1.81598   1.29780   1.11413  -1.90032  -0.08644   1.94707  -0.96654  -1.42437   1.73683   0.48509  -1.99917   0.59606   1.67682  -1.50289  -0.86406   1.97017  -0.20141  -1.86125   1.20797     ... ... ..... ....... ......... ........... .............
E33 *    2.00000  -0.65083  -1.57642   1.67682   0.48509  -1.99253   0.81172   1.46424  -1.76469  -0.31572   1.97017  -0.96654  -1.34111   1.83938   0.14399  -1.93309   1.11413   1.20797  -1.90032   0.02882   1.88157  -1.25341  -1.06581   1.94707  -0.20141  -1.81598   1.38332   0.91568  -1.97927   0.37250   1.73683  -1.50289  -0.75870   1.99668  -0.54080  -1.64471   1.61123   0.59606  -1.99917   0.70506   1.54029  -1.70754  -0.42897   1.98672  -0.86406  -1.42437   1.79108   0.25867  -1.95943   1.01659   1.29780  -1.86125  -0.08644   1.91750  -1.16154     ... ... ..... ....... ......... ........... .............
E34 *    2.00000  -0.75870  -1.42437   1.83938   0.02882  -1.86125   1.38332   0.81172  -1.99917   0.70506   1.46424  -1.81598  -0.08644   1.88157  -1.34111  -0.86406   1.99668  -0.65083  -1.50289   1.79108   0.14399  -1.90032   1.29780   0.91568  -1.99253   0.59606   1.54029  -1.76469  -0.20141   1.91750  -1.25341  -0.96654   1.98672  -0.54080  -1.57642   1.73683   0.25867  -1.93309   1.20797   1.01659  -1.97927   0.48509   1.61123  -1.70754  -0.31572   1.94707  -1.16154  -1.06581   1.97017  -0.42897  -1.64471   1.67682   0.37250  -1.95943   1.11413     ... ... ..... ....... ......... ........... .............
E35 *    2.00000  -0.86406  -1.25341   1.94707  -0.42897  -1.57642   1.79108   0.02882  -1.81598   1.54029   0.48509  -1.95943   1.20797   0.91568  -1.99917   0.81172   1.29780  -1.93309   0.37250   1.61123  -1.76469  -0.08644   1.83938  -1.50289  -0.54080   1.97017  -1.16154  -0.96654   1.99668  -0.75870  -1.34111   1.91750  -0.31572  -1.64471   1.73683   0.14399  -1.86125   1.46424   0.59606  -1.97927   1.11413   1.01659  -1.99253   0.70506   1.38332  -1.90032   0.25867   1.67682  -1.70754  -0.20141   1.88157  -1.42437  -0.65083   1.98672  -1.06581     ... ... ..... ....... ......... ........... .............
E36 *    2.00000  -0.96654  -1.06581   1.99668  -0.86406  -1.16154   1.98672  -0.75870  -1.25341   1.97017  -0.65083  -1.34111   1.94707  -0.54080  -1.42437   1.91750  -0.42897  -1.50289   1.88157  -0.31572  -1.57642   1.83938  -0.20141  -1.64471   1.79108  -0.08644  -1.70754   1.73683   0.02882  -1.76469   1.67682   0.14399  -1.81598   1.61123   0.25867  -1.86125   1.54029   0.37250  -1.90032   1.46424   0.48509  -1.93309   1.38332   0.59606  -1.95943   1.29780   0.70506  -1.97927   1.20797   0.81172  -1.99253   1.11413   0.91568  -1.99917   1.01659     ... ... ..... ....... ......... ........... .............
E37 *    2.00000  -1.06581  -0.86406   1.98672  -1.25341  -0.65083   1.94707  -1.42437  -0.42897   1.88157  -1.57642  -0.20141   1.79108  -1.70754   0.02882   1.67682  -1.81598   0.25867   1.54029  -1.90032   0.48509   1.38332  -1.95943   0.70506   1.20797  -1.99253   0.91568   1.01659  -1.99917   1.11413   0.81172  -1.97927   1.29780   0.59606  -1.93309   1.46424   0.37250  -1.86125   1.61123   0.14399  -1.76469   1.73683  -0.08644  -1.64471   1.83938  -0.31572  -1.50289   1.91750  -0.54080  -1.34111   1.97017  -0.75870  -1.16154   1.99668  -0.96654     ... ... ..... ....... ......... ........... .............
E38 *    2.00000  -1.16154  -0.65083   1.91750  -1.57642  -0.08644   1.67682  -1.86125   0.48509   1.29780  -1.99253   1.01659   0.81172  -1.95943   1.46424   0.25867  -1.76469   1.79108  -0.31572  -1.42437   1.97017  -0.86406  -0.96654   1.98672  -1.34111  -0.42897   1.83938  -1.70754   0.14399   1.54029  -1.93309   0.70506   1.11413  -1.99917   1.20797   0.59606  -1.90032   1.61123   0.02882  -1.64471   1.88157  -0.54080  -1.25341   1.99668  -1.06581  -0.75870   1.94707  -1.50289  -0.20141   1.73683  -1.81598   0.37250   1.38332  -1.97927   0.91568     ... ... ..... ....... ......... ........... .............
E39 *    2.00000  -1.25341  -0.42897   1.79108  -1.81598   0.48509   1.20797  -1.99917   1.29780   0.37250  -1.76469   1.83938  -0.54080  -1.16154   1.99668  -1.34111  -0.31572   1.73683  -1.86125   0.59606   1.11413  -1.99253   1.38332   0.25867  -1.70754   1.88157  -0.65083  -1.06581   1.98672  -1.42437  -0.20141   1.67682  -1.90032   0.70506   1.01659  -1.97927   1.46424   0.14399  -1.64471   1.91750  -0.75870  -0.96654   1.97017  -1.50289  -0.08644   1.61123  -1.93309   0.81172   0.91568  -1.95943   1.54029   0.02882  -1.57642   1.94707  -0.86406     ... ... ..... ....... ......... ........... .............
E40 *    2.00000  -1.34111  -0.20141   1.61123  -1.95943   1.01659   0.59606  -1.81598   1.83938  -0.65083  -0.96654   1.94707  -1.64471   0.25867   1.29780  -1.99917   1.38332   0.14399  -1.57642   1.97017  -1.06581  -0.54080   1.79108  -1.86125   0.70506   0.91568  -1.93309   1.67682  -0.31572  -1.25341   1.99668  -1.42437  -0.08644   1.54029  -1.97927   1.11413   0.48509  -1.76469   1.88157  -0.75870  -0.86406   1.91750  -1.70754   0.37250   1.20797  -1.99253   1.46424   0.02882  -1.50289   1.98672  -1.16154  -0.42897   1.73683  -1.90032   0.81172     ... ... ..... ....... ......... ........... .............
E41 *    2.00000  -1.42437   0.02882   1.38332  -1.99917   1.46424  -0.08644  -1.34111   1.99668  -1.50289   0.14399   1.29780  -1.99253   1.54029  -0.20141  -1.25341   1.98672  -1.57642   0.25867   1.20797  -1.97927   1.61123  -0.31572  -1.16154   1.97017  -1.64471   0.37250   1.11413  -1.95943   1.67682  -0.42897  -1.06581   1.94707  -1.70754   0.48509   1.01659  -1.93309   1.73683  -0.54080  -0.96654   1.91750  -1.76469   0.59606   0.91568  -1.90032   1.79108  -0.65083  -0.86406   1.88157  -1.81598   0.70506   0.81172  -1.86125   1.83938  -0.75870     ... ... ..... ....... ......... ........... .............
E42 *    2.00000  -1.50289   0.25867   1.11413  -1.93309   1.79108  -0.75870  -0.65083   1.73683  -1.95943   1.20797   0.14399  -1.42437   1.99668  -1.57642   0.37250   1.01659  -1.90032   1.83938  -0.86406  -0.54080   1.67682  -1.97927   1.29780   0.02882  -1.34111   1.98672  -1.64471   0.48509   0.91568  -1.86125   1.88157  -0.96654  -0.42897   1.61123  -1.99253   1.38332  -0.08644  -1.25341   1.97017  -1.70754   0.59606   0.81172  -1.81598   1.91750  -1.06581  -0.31572   1.54029  -1.99917   1.46424  -0.20141  -1.16154   1.94707  -1.76469   0.70506     ... ... ..... ....... ......... ........... .............
E43 *    2.00000  -1.57642   0.48509   0.81172  -1.76469   1.97017  -1.34111   0.14399   1.11413  -1.90032   1.88157  -1.06581  -0.20141   1.38332  -1.97927   1.73683  -0.75870  -0.54080   1.61123  -1.99917   1.54029  -0.42897  -0.86406   1.79108  -1.95943   1.29780  -0.08644  -1.16154   1.91750  -1.86125   1.01659   0.25867  -1.42437   1.98672  -1.70754   0.70506   0.59606  -1.64471   1.99668  -1.50289   0.37250   0.91568  -1.81598   1.94707  -1.25341   0.02882   1.20797  -1.93309   1.83938  -0.96654  -0.31572   1.46424  -1.99253   1.67682  -0.65083     ... ... ..... ....... ......... ........... .............
E44 *    2.00000  -1.64471   0.70506   0.48509  -1.50289   1.98672  -1.76469   0.91568   0.25867  -1.34111   1.94707  -1.86125   1.11413   0.02882  -1.16154   1.88157  -1.93309   1.29780  -0.20141  -0.96654   1.79108  -1.97927   1.46424  -0.42897  -0.75870   1.67682  -1.99917   1.61123  -0.65083  -0.54080   1.54029  -1.99253   1.73683  -0.86406  -0.31572   1.38332  -1.95943   1.83938  -1.06581  -0.08644   1.20797  -1.90032   1.91750  -1.25341   0.14399   1.01659  -1.81598   1.97017  -1.42437   0.37250   0.81172  -1.70754   1.99668  -1.57642   0.59606     ... ... ..... ....... ......... ........... .............
E45 *    2.00000  -1.70754   0.91568   0.14399  -1.16154   1.83938  -1.97927   1.54029  -0.65083  -0.42897   1.38332  -1.93309   1.91750  -1.34111   0.37250   0.70506  -1.57642   1.98672  -1.81598   1.11413  -0.08644  -0.96654   1.73683  -1.99917   1.67682  -0.86406  -0.20141   1.20797  -1.86125   1.97017  -1.50289   0.59606   0.48509  -1.42437   1.94707  -1.90032   1.29780  -0.31572  -0.75870   1.61123  -1.99253   1.79108  -1.06581   0.02882   1.01659  -1.76469   1.99668  -1.64471   0.81172   0.25867  -1.25341   1.88157  -1.95943   1.46424  -0.54080     ... ... ..... ....... ......... ........... .............
E46 *    2.00000  -1.76469   1.11413  -0.20141  -0.75870   1.54029  -1.95943   1.91750  -1.42437   0.59606   0.37250  -1.25341   1.83938  -1.99253   1.67682  -0.96654   0.02882   0.91568  -1.64471   1.98672  -1.86125   1.29780  -0.42897  -0.54080   1.38332  -1.90032   1.97017  -1.57642   0.81172   0.14399  -1.06581   1.73683  -1.99917   1.79108  -1.16154   0.25867   0.70506  -1.50289   1.94707  -1.93309   1.46424  -0.65083  -0.31572   1.20797  -1.81598   1.99668  -1.70754   1.01659  -0.08644  -0.86406   1.61123  -1.97927   1.88157  -1.34111   0.48509     ... ... ..... ....... ......... ........... .............
E47 *    2.00000  -1.81598   1.29780  -0.54080  -0.31572   1.11413  -1.70754   1.98672  -1.90032   1.46424  -0.75870  -0.08644   0.91568  -1.57642   1.94707  -1.95943   1.61123  -0.96654   0.14399   0.70506  -1.42437   1.88157  -1.99253   1.73683  -1.16154   0.37250   0.48509  -1.25341   1.79108  -1.99917   1.83938  -1.34111   0.59606   0.25867  -1.06581   1.67682  -1.97927   1.91750  -1.50289   0.81172   0.02882  -0.86406   1.54029  -1.93309   1.97017  -1.64471   1.01659  -0.20141  -0.65083   1.38332  -1.86125   1.99668  -1.76469   1.20797  -0.42897     ... ... ..... ....... ......... ........... .............
E48 *    2.00000  -1.86125   1.46424  -0.86406   0.14399   0.59606  -1.25341   1.73683  -1.97927   1.94707  -1.64471   1.11413  -0.42897  -0.31572   1.01659  -1.57642   1.91750  -1.99253   1.79108  -1.34111   0.70506   0.02882  -0.75870   1.38332  -1.81598   1.99668  -1.90032   1.54029  -0.96654   0.25867   0.48509  -1.16154   1.67682  -1.95943   1.97017  -1.70754   1.20797  -0.54080  -0.20141   0.91568  -1.50289   1.88157  -1.99917   1.83938  -1.42437   0.81172  -0.08644  -0.65083   1.29780  -1.76469   1.98672  -1.93309   1.61123  -1.06581   0.37250     ... ... ..... ....... ......... ........... .............
E49 *    2.00000  -1.90032   1.61123  -1.16154   0.59606   0.02882  -0.65083   1.20797  -1.64471   1.91750  -1.99917   1.88157  -1.57642   1.11413  -0.54080  -0.08644   0.70506  -1.25341   1.67682  -1.93309   1.99668  -1.86125   1.54029  -1.06581   0.48509   0.14399  -0.75870   1.29780  -1.70754   1.94707  -1.99253   1.83938  -1.50289   1.01659  -0.42897  -0.20141   0.81172  -1.34111   1.73683  -1.95943   1.98672  -1.81598   1.46424  -0.96654   0.37250   0.25867  -0.86406   1.38332  -1.76469   1.97017  -1.97927   1.79108  -1.42437   0.91568  -0.31572     ... ... ..... ....... ......... ........... .............
E50 *    2.00000  -1.93309   1.73683  -1.42437   1.01659  -0.54080   0.02882   0.48509  -0.96654   1.38332  -1.70754   1.91750  -1.99917   1.94707  -1.76469   1.46424  -1.06581   0.59606  -0.08644  -0.42897   0.91568  -1.34111   1.67682  -1.90032   1.99668  -1.95943   1.79108  -1.50289   1.11413  -0.65083   0.14399   0.37250  -0.86406   1.29780  -1.64471   1.88157  -1.99253   1.97017  -1.81598   1.54029  -1.16154   0.70506  -0.20141  -0.31572   0.81172  -1.25341   1.61123  -1.86125   1.98672  -1.97927   1.83938  -1.57642   1.20797  -0.75870   0.25867     ... ... ..... ....... ......... ........... .............
E51 *    2.00000  -1.95943   1.83938  -1.64471   1.38332  -1.06581   0.70506  -0.31572  -0.08644   0.48509  -0.86406   1.20797  -1.50289   1.73683  -1.90032   1.98672  -1.99253   1.91750  -1.76469   1.54029  -1.25341   0.91568  -0.54080   0.14399   0.25867  -0.65083   1.01659  -1.34111   1.61123  -1.81598   1.94707  -1.99917   1.97017  -1.86125   1.67682  -1.42437   1.11413  -0.75870   0.37250   0.02882  -0.42897   0.81172  -1.16154   1.46424  -1.70754   1.88157  -1.97927   1.99668  -1.93309   1.79108  -1.57642   1.29780  -0.96654   0.59606  -0.20141     ... ... ..... ....... ......... ........... .............
E52 *    2.00000  -1.97927   1.91750  -1.81598   1.67682  -1.50289   1.29780  -1.06581   0.81172  -0.54080   0.25867   0.02882  -0.31572   0.59606  -0.86406   1.11413  -1.34111   1.54029  -1.70754   1.83938  -1.93309   1.98672  -1.99917   1.97017  -1.90032   1.79108  -1.64471   1.46424  -1.25341   1.01659  -0.75870   0.48509  -0.20141  -0.08644   0.37250  -0.65083   0.91568  -1.16154   1.38332  -1.57642   1.73683  -1.86125   1.94707  -1.99253   1.99668  -1.95943   1.88157  -1.76469   1.61123  -1.42437   1.20797  -0.96654   0.70506  -0.42897   0.14399     ... ... ..... ....... ......... ........... .............
E53 *    2.00000  -1.99253   1.97017  -1.93309   1.88157  -1.81598   1.73683  -1.64471   1.54029  -1.42437   1.29780  -1.16154   1.01659  -0.86406   0.70506  -0.54080   0.37250  -0.20141   0.02882   0.14399  -0.31572   0.48509  -0.65083   0.81172  -0.96654   1.11413  -1.25341   1.38332  -1.50289   1.61123  -1.70754   1.79108  -1.86125   1.91750  -1.95943   1.98672  -1.99917   1.99668  -1.97927   1.94707  -1.90032   1.83938  -1.76469   1.67682  -1.57642   1.46424  -1.34111   1.20797  -1.06581   0.91568  -0.75870   0.59606  -0.42897   0.25867  -0.08644     ... ... ..... ....... ......... ........... .............
E54 *    2.00000  -1.99917   1.99668  -1.99253   1.98672  -1.97927   1.97017  -1.95943   1.94707  -1.93309   1.91750  -1.90032   1.88157  -1.86125   1.83938  -1.81598   1.79108  -1.76469   1.73683  -1.70754   1.67682  -1.64471   1.61123  -1.57642   1.54029  -1.50289   1.46424  -1.42437   1.38332  -1.34111   1.29780  -1.25341   1.20797  -1.16154   1.11413  -1.06581   1.01659  -0.96654   0.91568  -0.86406   0.81172  -0.75870   0.70506  -0.65083   0.59606  -0.54080   0.48509  -0.42897   0.37250  -0.31572   0.25867  -0.20141   0.14399  -0.08644   0.02882     ... ... ..... ....... ......... ........... .............

 Irrational character values:  1.996678100566 = 2*cos(2*π/109)
                               1.986723437279 = 2*cos(4*π/109)
                               1.970169078530 = 2*cos(6*π/109)
                               1.947070016234 = 2*cos(8*π/109)
                               1.917502983153 = 2*cos(10*π/109)
                               1.881566197997 = 2*cos(12*π/109)
                               1.839379039153 = 2*cos(14*π/109)
                               1.791081648119 = 2*cos(16*π/109)
                               1.736834463972 = 2*cos(18*π/109)
                               1.676817690402 = 2*cos(20*π/109)
                               1.611230697094 = 2*cos(22*π/109)
                               1.540291357446 = 2*cos(24*π/109)
                               1.464235324809 = 2*cos(26*π/109)
                               1.383315249675 = 2*cos(28*π/109)
                               1.297799940396 = 2*cos(30*π/109)
                               1.207973470229 = 2*cos(32*π/109)
                               1.114134233675 = 2*cos(34*π/109)
                               1.016593955240 = 2*cos(36*π/109)
                               0.915676653921 = 2*cos(38*π/109)
                               0.811717566843 = 2*cos(40*π/109)
                               0.705062035640 = 2*cos(42*π/109)
                               0.596064359259 = 2*cos(44*π/109)
                               0.485086617020 = 2*cos(46*π/109)
                               0.372497465823 = 2*cos(48*π/109)
                               0.258670915504 = 2*cos(50*π/109)
                               0.143985086418 = 2*cos(52*π/109)
                               0.028820953355 = 2*cos(54*π/109)
                              -0.086438920017 = 2*cos(56*π/109)
                              -0.201411651989 = 2*cos(58*π/109)
                              -0.315715314709 = 2*cos(60*π/109)
                              -0.428970202903 = 2*cos(62*π/109)
                              -0.540800095223 = 2*cos(64*π/109)
                              -0.650833504013 = 2*cos(66*π/109)
                              -0.758704909354 = 2*cos(68*π/109)
                              -0.864055973286 = 2*cos(70*π/109)
                              -0.966536730169 = 2*cos(72*π/109)
                              -1.065806749235 = 2*cos(74*π/109)
                              -1.161536265464 = 2*cos(76*π/109)
                              -1.253407275030 = 2*cos(78*π/109)
                              -1.341114591678 = 2*cos(80*π/109)
                              -1.424366860523 = 2*cos(82*π/109)
                              -1.502887525900 = 2*cos(84*π/109)
                              -1.576415750055 = 2*cos(86*π/109)
                              -1.644707279621 = 2*cos(88*π/109)
                              -1.707535257007 = 2*cos(90*π/109)
                              -1.764690973988 = 2*cos(92*π/109)
                              -1.815984565021 = 2*cos(94*π/109)
                              -1.861245637956 = 2*cos(96*π/109)
                              -1.900323840058 = 2*cos(98*π/109)
                              -1.933089357472 = 2*cos(100*π/109)
                              -1.959433346443 = 2*cos(102*π/109)
                              -1.979268294889 = 2*cos(104*π/109)
                              -1.992528313106 = 2*cos(106*π/109)
                              -1.999169352648 = 2*cos(108*π/109)



 Symmetry of Rotations and Cartesian products

A    R+p+d+f+g+h+i+j+k+l+m  Rz, z, z2, z3, z4, z5, z6 
E1   R+p+d+f+g+h+i+j+k+l+m  {Rx, Ry}, {x, y}, {xz, yz}, {xz2, yz2}, {xz3, yz3}, {xz4, yz4}, {xz5, yz5} 
E2   d+f+g+h+i+j+k+l+m      {x2y2, xy}, {z(x2y2), xyz}, {z2(x2y2), xyz2}, {z3(x2y2), xyz3}, {z4(x2y2), xyz4} 
E3   f+g+h+i+j+k+l+m        {x(x2−3y2), y(3x2y2)}, {xz(x2−3y2), yz(3x2y2)}, {xz2(x2−3y2), yz2(3x2y2)}, {xz3(x2−3y2), yz3(3x2y2)} 
E4   g+h+i+j+k+l+m          {(x2y2)2−4x2y2, xy(x2y2)}, {z((x2y2)2−4x2y2), xyz(x2y2)}, {z2((x2y2)2−4x2y2), xyz2(x2y2)} 
E5   h+i+j+k+l+m            {x(x2−(5+2√5)y2)(x2−(5−2√5)y2), y((5+2√5)x2y2)((5−2√5)x2y2)}, {xz(x2−(5+2√5)y2)(x2−(5−2√5)y2), yz((5+2√5)x2y2)((5−2√5)x2y2)} 
E6   i+j+k+l+m              {x2(x2−3y2)2y2(3x2y2)2, xy(x2−3y2)(3x2y2)} 
E7   j+k+l+m 
E8   k+l+m 
E9   l+m 
E10  m 

 Notes:

    α  The order of the C109 point group is 109, and the order of the principal axis (C109) is 109. The group has 55 irreducible representations.

    β  The C109 point group is generated by one single symmetry element, C109. Therefore, it is a cyclic group.

    γ  The lowest nonvanishing multipole moment in C109 is 2 (dipole moment).

    δ  This is an Abelian point group (the commutative law holds between all symmetry operations).
       The C109 group is Abelian because it contains only one symmetry element, all the powers of which necessarily commute (sufficient condition).
       In Abelian groups, all symmetry operations form a class of their own, and all irreducible representations are one-dimensional.

    ε  Because the group is Abelian and the maximum order of rotation is >2, some irreducible representations have complex characters.
       These 108 cases have been combined into 54 two-dimensional representations that are no longer irreducible but have real-valued characters.
       Accordingly, 54 pairs of left and right rotations have been combined into one two-membered pseudo-class each.

    ζ  The 54 reducible “E” representations almost behave like true irreducible representations.
       Their norm, however, is twice the group order. Therefore, they have been marked with an asterisk in the table.
       This is essential when trying to decompose a reducible representation into “irreducible” ones using the familiar projection formula.

    η  The point group is chiral, as it does not contain any mirroring operation.

    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 just less than half the order of the principal axis.

    ι  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 C109 point group was created by Gernot Katzer.

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