Character table for the S112 point group

S112    E         2 S112    2 C56     2 S112^3  2 C28     2 S112^5  2 C56^3   2 S16     2 C14     2 S112^9  2 C56^5   2 S112^11 2 C28^3   2 S112^13 2 C8      2 S112^15 2 C7      2 S112^17 2 C56^9   2 S112^19 2 C28^5   2 S16^3   2 C56^11  2 S112^23 2 C14^3   2 S112^25 2 C56^13  2 S112^27 2 C4      2 S112^29 2 C56^15  2 S112^31 2 C7^2    2 S112^33 2 C56^17  2 S16^5   2 C28^9   2 S112^37 2 C56^19  2 S112^39 2 C14^5   2 S112^41 2 C8^3    2 S112^43 2 C28^11  2 S112^45 2 C56^23  2 S112^47 2 C7^3    2 S16^7   2 C56^25  2 S112^51 2 C28^13  2 S112^53 2 C56^27  2 S112^55 C2           <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   1.00000   1.00000     ..T ... ....T ....... ........T ........... ............T
B        1.00000  -1.00000   1.00000  -1.00000   1.00000  -1.00000   1.00000  -1.00000   1.00000  -1.00000   1.00000  -1.00000   1.00000  -1.00000   1.00000  -1.00000   1.00000  -1.00000   1.00000  -1.00000   1.00000  -1.00000   1.00000  -1.00000   1.00000  -1.00000   1.00000  -1.00000   1.00000  -1.00000   1.00000  -1.00000   1.00000  -1.00000   1.00000  -1.00000   1.00000  -1.00000   1.00000  -1.00000   1.00000  -1.00000   1.00000  -1.00000   1.00000  -1.00000   1.00000  -1.00000   1.00000  -1.00000   1.00000  -1.00000   1.00000  -1.00000   1.00000  -1.00000   1.00000     ... ..T ..... ......T ......... ..........T .............
E1  *    2.00000   1.99685   1.98742   1.97174   1.94986   1.92183   1.88777   1.84776   1.80194   1.75045   1.69345   1.63112   1.56366   1.49128   1.41421   1.33269   1.24698   1.15734   1.06406   0.96744   0.86777   0.76537   0.66056   0.55367   0.44504   0.33501   0.22393   0.11214   0.00000  -0.11214  -0.22393  -0.33501  -0.44504  -0.55367  -0.66056  -0.76537  -0.86777  -0.96744  -1.06406  -1.15734  -1.24698  -1.33269  -1.41421  -1.49128  -1.56366  -1.63112  -1.69345  -1.75045  -1.80194  -1.84776  -1.88777  -1.92183  -1.94986  -1.97174  -1.98742  -1.99685  -2.00000     ... TT. ..... ....TT. ......... ........TT. .............
E2  *    2.00000   1.98742   1.94986   1.88777   1.80194   1.69345   1.56366   1.41421   1.24698   1.06406   0.86777   0.66056   0.44504   0.22393   0.00000  -0.22393  -0.44504  -0.66056  -0.86777  -1.06406  -1.24698  -1.41421  -1.56366  -1.69345  -1.80194  -1.88777  -1.94986  -1.98742  -2.00000  -1.98742  -1.94986  -1.88777  -1.80194  -1.69345  -1.56366  -1.41421  -1.24698  -1.06406  -0.86777  -0.66056  -0.44504  -0.22393   0.00000   0.22393   0.44504   0.66056   0.86777   1.06406   1.24698   1.41421   1.56366   1.69345   1.80194   1.88777   1.94986   1.98742   2.00000     ... ... TT... ....... ....TT... ........... ........TT...
E3  *    2.00000   1.97174   1.88777   1.75045   1.56366   1.33269   1.06406   0.76537   0.44504   0.11214  -0.22393  -0.55367  -0.86777  -1.15734  -1.41421  -1.63112  -1.80194  -1.92183  -1.98742  -1.99685  -1.94986  -1.84776  -1.69345  -1.49128  -1.24698  -0.96744  -0.66056  -0.33501   0.00000   0.33501   0.66056   0.96744   1.24698   1.49128   1.69345   1.84776   1.94986   1.99685   1.98742   1.92183   1.80194   1.63112   1.41421   1.15734   0.86777   0.55367   0.22393  -0.11214  -0.44504  -0.76537  -1.06406  -1.33269  -1.56366  -1.75045  -1.88777  -1.97174  -2.00000     ... ... ..... TT..... ......... ....TT..... .............
E4  *    2.00000   1.94986   1.80194   1.56366   1.24698   0.86777   0.44504   0.00000  -0.44504  -0.86777  -1.24698  -1.56366  -1.80194  -1.94986  -2.00000  -1.94986  -1.80194  -1.56366  -1.24698  -0.86777  -0.44504   0.00000   0.44504   0.86777   1.24698   1.56366   1.80194   1.94986   2.00000   1.94986   1.80194   1.56366   1.24698   0.86777   0.44504   0.00000  -0.44504  -0.86777  -1.24698  -1.56366  -1.80194  -1.94986  -2.00000  -1.94986  -1.80194  -1.56366  -1.24698  -0.86777  -0.44504   0.00000   0.44504   0.86777   1.24698   1.56366   1.80194   1.94986   2.00000     ... ... ..... ....... TT....... ........... ....TT.......
E5  *    2.00000   1.92183   1.69345   1.33269   0.86777   0.33501  -0.22393  -0.76537  -1.24698  -1.63112  -1.88777  -1.99685  -1.94986  -1.75045  -1.41421  -0.96744  -0.44504   0.11214   0.66056   1.15734   1.56366   1.84776   1.98742   1.97174   1.80194   1.49128   1.06406   0.55367   0.00000  -0.55367  -1.06406  -1.49128  -1.80194  -1.97174  -1.98742  -1.84776  -1.56366  -1.15734  -0.66056  -0.11214   0.44504   0.96744   1.41421   1.75045   1.94986   1.99685   1.88777   1.63112   1.24698   0.76537   0.22393  -0.33501  -0.86777  -1.33269  -1.69345  -1.92183  -2.00000     ... ... ..... ....... ......... TT......... .............
E6  *    2.00000   1.88777   1.56366   1.06406   0.44504  -0.22393  -0.86777  -1.41421  -1.80194  -1.98742  -1.94986  -1.69345  -1.24698  -0.66056   0.00000   0.66056   1.24698   1.69345   1.94986   1.98742   1.80194   1.41421   0.86777   0.22393  -0.44504  -1.06406  -1.56366  -1.88777  -2.00000  -1.88777  -1.56366  -1.06406  -0.44504   0.22393   0.86777   1.41421   1.80194   1.98742   1.94986   1.69345   1.24698   0.66056   0.00000  -0.66056  -1.24698  -1.69345  -1.94986  -1.98742  -1.80194  -1.41421  -0.86777  -0.22393   0.44504   1.06406   1.56366   1.88777   2.00000     ... ... ..... ....... ......... ........... TT...........
E7  *    2.00000   1.84776   1.41421   0.76537   0.00000  -0.76537  -1.41421  -1.84776  -2.00000  -1.84776  -1.41421  -0.76537   0.00000   0.76537   1.41421   1.84776   2.00000   1.84776   1.41421   0.76537   0.00000  -0.76537  -1.41421  -1.84776  -2.00000  -1.84776  -1.41421  -0.76537   0.00000   0.76537   1.41421   1.84776   2.00000   1.84776   1.41421   0.76537   0.00000  -0.76537  -1.41421  -1.84776  -2.00000  -1.84776  -1.41421  -0.76537   0.00000   0.76537   1.41421   1.84776   2.00000   1.84776   1.41421   0.76537   0.00000  -0.76537  -1.41421  -1.84776  -2.00000     ... ... ..... ....... ......... ........... .............
E8  *    2.00000   1.80194   1.24698   0.44504  -0.44504  -1.24698  -1.80194  -2.00000  -1.80194  -1.24698  -0.44504   0.44504   1.24698   1.80194   2.00000   1.80194   1.24698   0.44504  -0.44504  -1.24698  -1.80194  -2.00000  -1.80194  -1.24698  -0.44504   0.44504   1.24698   1.80194   2.00000   1.80194   1.24698   0.44504  -0.44504  -1.24698  -1.80194  -2.00000  -1.80194  -1.24698  -0.44504   0.44504   1.24698   1.80194   2.00000   1.80194   1.24698   0.44504  -0.44504  -1.24698  -1.80194  -2.00000  -1.80194  -1.24698  -0.44504   0.44504   1.24698   1.80194   2.00000     ... ... ..... ....... ......... ........... .............
E9  *    2.00000   1.75045   1.06406   0.11214  -0.86777  -1.63112  -1.98742  -1.84776  -1.24698  -0.33501   0.66056   1.49128   1.94986   1.92183   1.41421   0.55367  -0.44504  -1.33269  -1.88777  -1.97174  -1.56366  -0.76537   0.22393   1.15734   1.80194   1.99685   1.69345   0.96744   0.00000  -0.96744  -1.69345  -1.99685  -1.80194  -1.15734  -0.22393   0.76537   1.56366   1.97174   1.88777   1.33269   0.44504  -0.55367  -1.41421  -1.92183  -1.94986  -1.49128  -0.66056   0.33501   1.24698   1.84776   1.98742   1.63112   0.86777  -0.11214  -1.06406  -1.75045  -2.00000     ... ... ..... ....... ......... ........... .............
E10 *    2.00000   1.69345   0.86777  -0.22393  -1.24698  -1.88777  -1.94986  -1.41421  -0.44504   0.66056   1.56366   1.98742   1.80194   1.06406   0.00000  -1.06406  -1.80194  -1.98742  -1.56366  -0.66056   0.44504   1.41421   1.94986   1.88777   1.24698   0.22393  -0.86777  -1.69345  -2.00000  -1.69345  -0.86777   0.22393   1.24698   1.88777   1.94986   1.41421   0.44504  -0.66056  -1.56366  -1.98742  -1.80194  -1.06406   0.00000   1.06406   1.80194   1.98742   1.56366   0.66056  -0.44504  -1.41421  -1.94986  -1.88777  -1.24698  -0.22393   0.86777   1.69345   2.00000     ... ... ..... ....... ......... ........... .............
E11 *    2.00000   1.63112   0.66056  -0.55367  -1.56366  -1.99685  -1.69345  -0.76537   0.44504   1.49128   1.98742   1.75045   0.86777  -0.33501  -1.41421  -1.97174  -1.80194  -0.96744   0.22393   1.33269   1.94986   1.84776   1.06406  -0.11214  -1.24698  -1.92183  -1.88777  -1.15734   0.00000   1.15734   1.88777   1.92183   1.24698   0.11214  -1.06406  -1.84776  -1.94986  -1.33269  -0.22393   0.96744   1.80194   1.97174   1.41421   0.33501  -0.86777  -1.75045  -1.98742  -1.49128  -0.44504   0.76537   1.69345   1.99685   1.56366   0.55367  -0.66056  -1.63112  -2.00000     ... ... ..... ....... ......... ........... .............
E12 *    2.00000   1.56366   0.44504  -0.86777  -1.80194  -1.94986  -1.24698   0.00000   1.24698   1.94986   1.80194   0.86777  -0.44504  -1.56366  -2.00000  -1.56366  -0.44504   0.86777   1.80194   1.94986   1.24698   0.00000  -1.24698  -1.94986  -1.80194  -0.86777   0.44504   1.56366   2.00000   1.56366   0.44504  -0.86777  -1.80194  -1.94986  -1.24698   0.00000   1.24698   1.94986   1.80194   0.86777  -0.44504  -1.56366  -2.00000  -1.56366  -0.44504   0.86777   1.80194   1.94986   1.24698   0.00000  -1.24698  -1.94986  -1.80194  -0.86777   0.44504   1.56366   2.00000     ... ... ..... ....... ......... ........... .............
E13 *    2.00000   1.49128   0.22393  -1.15734  -1.94986  -1.75045  -0.66056   0.76537   1.80194   1.92183   1.06406  -0.33501  -1.56366  -1.99685  -1.41421  -0.11214   1.24698   1.97174   1.69345   0.55367  -0.86777  -1.84776  -1.88777  -0.96744   0.44504   1.63112   1.98742   1.33269   0.00000  -1.33269  -1.98742  -1.63112  -0.44504   0.96744   1.88777   1.84776   0.86777  -0.55367  -1.69345  -1.97174  -1.24698   0.11214   1.41421   1.99685   1.56366   0.33501  -1.06406  -1.92183  -1.80194  -0.76537   0.66056   1.75045   1.94986   1.15734  -0.22393  -1.49128  -2.00000     ... ... ..... ....... ......... ........... .............
E14 *    2.00000   1.41421   0.00000  -1.41421  -2.00000  -1.41421   0.00000   1.41421   2.00000   1.41421   0.00000  -1.41421  -2.00000  -1.41421   0.00000   1.41421   2.00000   1.41421   0.00000  -1.41421  -2.00000  -1.41421   0.00000   1.41421   2.00000   1.41421   0.00000  -1.41421  -2.00000  -1.41421   0.00000   1.41421   2.00000   1.41421   0.00000  -1.41421  -2.00000  -1.41421   0.00000   1.41421   2.00000   1.41421   0.00000  -1.41421  -2.00000  -1.41421   0.00000   1.41421   2.00000   1.41421   0.00000  -1.41421  -2.00000  -1.41421   0.00000   1.41421   2.00000     ... ... ..... ....... ......... ........... .............
E15 *    2.00000   1.33269  -0.22393  -1.63112  -1.94986  -0.96744   0.66056   1.84776   1.80194   0.55367  -1.06406  -1.97174  -1.56366  -0.11214   1.41421   1.99685   1.24698  -0.33501  -1.69345  -1.92183  -0.86777   0.76537   1.88777   1.75045   0.44504  -1.15734  -1.98742  -1.49128   0.00000   1.49128   1.98742   1.15734  -0.44504  -1.75045  -1.88777  -0.76537   0.86777   1.92183   1.69345   0.33501  -1.24698  -1.99685  -1.41421   0.11214   1.56366   1.97174   1.06406  -0.55367  -1.80194  -1.84776  -0.66056   0.96744   1.94986   1.63112   0.22393  -1.33269  -2.00000     ... ... ..... ....... ......... ........... .............
E16 *    2.00000   1.24698  -0.44504  -1.80194  -1.80194  -0.44504   1.24698   2.00000   1.24698  -0.44504  -1.80194  -1.80194  -0.44504   1.24698   2.00000   1.24698  -0.44504  -1.80194  -1.80194  -0.44504   1.24698   2.00000   1.24698  -0.44504  -1.80194  -1.80194  -0.44504   1.24698   2.00000   1.24698  -0.44504  -1.80194  -1.80194  -0.44504   1.24698   2.00000   1.24698  -0.44504  -1.80194  -1.80194  -0.44504   1.24698   2.00000   1.24698  -0.44504  -1.80194  -1.80194  -0.44504   1.24698   2.00000   1.24698  -0.44504  -1.80194  -1.80194  -0.44504   1.24698   2.00000     ... ... ..... ....... ......... ........... .............
E17 *    2.00000   1.15734  -0.66056  -1.92183  -1.56366   0.11214   1.69345   1.84776   0.44504  -1.33269  -1.98742  -0.96744   0.86777   1.97174   1.41421  -0.33501  -1.80194  -1.75045  -0.22393   1.49128   1.94986   0.76537  -1.06406  -1.99685  -1.24698   0.55367   1.88777   1.63112   0.00000  -1.63112  -1.88777  -0.55367   1.24698   1.99685   1.06406  -0.76537  -1.94986  -1.49128   0.22393   1.75045   1.80194   0.33501  -1.41421  -1.97174  -0.86777   0.96744   1.98742   1.33269  -0.44504  -1.84776  -1.69345  -0.11214   1.56366   1.92183   0.66056  -1.15734  -2.00000     ... ... ..... ....... ......... ........... .............
E18 *    2.00000   1.06406  -0.86777  -1.98742  -1.24698   0.66056   1.94986   1.41421  -0.44504  -1.88777  -1.56366   0.22393   1.80194   1.69345   0.00000  -1.69345  -1.80194  -0.22393   1.56366   1.88777   0.44504  -1.41421  -1.94986  -0.66056   1.24698   1.98742   0.86777  -1.06406  -2.00000  -1.06406   0.86777   1.98742   1.24698  -0.66056  -1.94986  -1.41421   0.44504   1.88777   1.56366  -0.22393  -1.80194  -1.69345   0.00000   1.69345   1.80194   0.22393  -1.56366  -1.88777  -0.44504   1.41421   1.94986   0.66056  -1.24698  -1.98742  -0.86777   1.06406   2.00000     ... ... ..... ....... ......... ........... .............
E19 *    2.00000   0.96744  -1.06406  -1.99685  -0.86777   1.15734   1.98742   0.76537  -1.24698  -1.97174  -0.66056   1.33269   1.94986   0.55367  -1.41421  -1.92183  -0.44504   1.49128   1.88777   0.33501  -1.56366  -1.84776  -0.22393   1.63112   1.80194   0.11214  -1.69345  -1.75045   0.00000   1.75045   1.69345  -0.11214  -1.80194  -1.63112   0.22393   1.84776   1.56366  -0.33501  -1.88777  -1.49128   0.44504   1.92183   1.41421  -0.55367  -1.94986  -1.33269   0.66056   1.97174   1.24698  -0.76537  -1.98742  -1.15734   0.86777   1.99685   1.06406  -0.96744  -2.00000     ... ... ..... ....... ......... ........... .............
E20 *    2.00000   0.86777  -1.24698  -1.94986  -0.44504   1.56366   1.80194   0.00000  -1.80194  -1.56366   0.44504   1.94986   1.24698  -0.86777  -2.00000  -0.86777   1.24698   1.94986   0.44504  -1.56366  -1.80194   0.00000   1.80194   1.56366  -0.44504  -1.94986  -1.24698   0.86777   2.00000   0.86777  -1.24698  -1.94986  -0.44504   1.56366   1.80194   0.00000  -1.80194  -1.56366   0.44504   1.94986   1.24698  -0.86777  -2.00000  -0.86777   1.24698   1.94986   0.44504  -1.56366  -1.80194   0.00000   1.80194   1.56366  -0.44504  -1.94986  -1.24698   0.86777   2.00000     ... ... ..... ....... ......... ........... .............
E21 *    2.00000   0.76537  -1.41421  -1.84776   0.00000   1.84776   1.41421  -0.76537  -2.00000  -0.76537   1.41421   1.84776   0.00000  -1.84776  -1.41421   0.76537   2.00000   0.76537  -1.41421  -1.84776   0.00000   1.84776   1.41421  -0.76537  -2.00000  -0.76537   1.41421   1.84776   0.00000  -1.84776  -1.41421   0.76537   2.00000   0.76537  -1.41421  -1.84776   0.00000   1.84776   1.41421  -0.76537  -2.00000  -0.76537   1.41421   1.84776   0.00000  -1.84776  -1.41421   0.76537   2.00000   0.76537  -1.41421  -1.84776   0.00000   1.84776   1.41421  -0.76537  -2.00000     ... ... ..... ....... ......... ........... .............
E22 *    2.00000   0.66056  -1.56366  -1.69345   0.44504   1.98742   0.86777  -1.41421  -1.80194   0.22393   1.94986   1.06406  -1.24698  -1.88777   0.00000   1.88777   1.24698  -1.06406  -1.94986  -0.22393   1.80194   1.41421  -0.86777  -1.98742  -0.44504   1.69345   1.56366  -0.66056  -2.00000  -0.66056   1.56366   1.69345  -0.44504  -1.98742  -0.86777   1.41421   1.80194  -0.22393  -1.94986  -1.06406   1.24698   1.88777   0.00000  -1.88777  -1.24698   1.06406   1.94986   0.22393  -1.80194  -1.41421   0.86777   1.98742   0.44504  -1.69345  -1.56366   0.66056   2.00000     ... ... ..... ....... ......... ........... .............
E23 *    2.00000   0.55367  -1.69345  -1.49128   0.86777   1.97174   0.22393  -1.84776  -1.24698   1.15734   1.88777  -0.11214  -1.94986  -0.96744   1.41421   1.75045  -0.44504  -1.99685  -0.66056   1.63112   1.56366  -0.76537  -1.98742  -0.33501   1.80194   1.33269  -1.06406  -1.92183   0.00000   1.92183   1.06406  -1.33269  -1.80194   0.33501   1.98742   0.76537  -1.56366  -1.63112   0.66056   1.99685   0.44504  -1.75045  -1.41421   0.96744   1.94986   0.11214  -1.88777  -1.15734   1.24698   1.84776  -0.22393  -1.97174  -0.86777   1.49128   1.69345  -0.55367  -2.00000     ... ... ..... ....... ......... ........... .............
E24 *    2.00000   0.44504  -1.80194  -1.24698   1.24698   1.80194  -0.44504  -2.00000  -0.44504   1.80194   1.24698  -1.24698  -1.80194   0.44504   2.00000   0.44504  -1.80194  -1.24698   1.24698   1.80194  -0.44504  -2.00000  -0.44504   1.80194   1.24698  -1.24698  -1.80194   0.44504   2.00000   0.44504  -1.80194  -1.24698   1.24698   1.80194  -0.44504  -2.00000  -0.44504   1.80194   1.24698  -1.24698  -1.80194   0.44504   2.00000   0.44504  -1.80194  -1.24698   1.24698   1.80194  -0.44504  -2.00000  -0.44504   1.80194   1.24698  -1.24698  -1.80194   0.44504   2.00000     ... ... ..... ....... ......... ........... .............
E25 *    2.00000   0.33501  -1.88777  -0.96744   1.56366   1.49128  -1.06406  -1.84776   0.44504   1.99685   0.22393  -1.92183  -0.86777   1.63112   1.41421  -1.15734  -1.80194   0.55367   1.98742   0.11214  -1.94986  -0.76537   1.69345   1.33269  -1.24698  -1.75045   0.66056   1.97174   0.00000  -1.97174  -0.66056   1.75045   1.24698  -1.33269  -1.69345   0.76537   1.94986  -0.11214  -1.98742  -0.55367   1.80194   1.15734  -1.41421  -1.63112   0.86777   1.92183  -0.22393  -1.99685  -0.44504   1.84776   1.06406  -1.49128  -1.56366   0.96744   1.88777  -0.33501  -2.00000     ... ... ..... ....... ......... ........... .............
E26 *    2.00000   0.22393  -1.94986  -0.66056   1.80194   1.06406  -1.56366  -1.41421   1.24698   1.69345  -0.86777  -1.88777   0.44504   1.98742   0.00000  -1.98742  -0.44504   1.88777   0.86777  -1.69345  -1.24698   1.41421   1.56366  -1.06406  -1.80194   0.66056   1.94986  -0.22393  -2.00000  -0.22393   1.94986   0.66056  -1.80194  -1.06406   1.56366   1.41421  -1.24698  -1.69345   0.86777   1.88777  -0.44504  -1.98742   0.00000   1.98742   0.44504  -1.88777  -0.86777   1.69345   1.24698  -1.41421  -1.56366   1.06406   1.80194  -0.66056  -1.94986   0.22393   2.00000     ... ... ..... ....... ......... ........... .............
E27 *    2.00000   0.11214  -1.98742  -0.33501   1.94986   0.55367  -1.88777  -0.76537   1.80194   0.96744  -1.69345  -1.15734   1.56366   1.33269  -1.41421  -1.49128   1.24698   1.63112  -1.06406  -1.75045   0.86777   1.84776  -0.66056  -1.92183   0.44504   1.97174  -0.22393  -1.99685   0.00000   1.99685   0.22393  -1.97174  -0.44504   1.92183   0.66056  -1.84776  -0.86777   1.75045   1.06406  -1.63112  -1.24698   1.49128   1.41421  -1.33269  -1.56366   1.15734   1.69345  -0.96744  -1.80194   0.76537   1.88777  -0.55367  -1.94986   0.33501   1.98742  -0.11214  -2.00000     ... ... ..... ....... ......... ........... .............
E28 *    2.00000   0.00000  -2.00000   0.00000   2.00000   0.00000  -2.00000   0.00000   2.00000   0.00000  -2.00000   0.00000   2.00000   0.00000  -2.00000   0.00000   2.00000   0.00000  -2.00000   0.00000   2.00000   0.00000  -2.00000   0.00000   2.00000   0.00000  -2.00000   0.00000   2.00000   0.00000  -2.00000   0.00000   2.00000   0.00000  -2.00000   0.00000   2.00000   0.00000  -2.00000   0.00000   2.00000   0.00000  -2.00000   0.00000   2.00000   0.00000  -2.00000   0.00000   2.00000   0.00000  -2.00000   0.00000   2.00000   0.00000  -2.00000   0.00000   2.00000     ... ... ..... ....... ......... ........... .............
E29 *    2.00000  -0.11214  -1.98742   0.33501   1.94986  -0.55367  -1.88777   0.76537   1.80194  -0.96744  -1.69345   1.15734   1.56366  -1.33269  -1.41421   1.49128   1.24698  -1.63112  -1.06406   1.75045   0.86777  -1.84776  -0.66056   1.92183   0.44504  -1.97174  -0.22393   1.99685   0.00000  -1.99685   0.22393   1.97174  -0.44504  -1.92183   0.66056   1.84776  -0.86777  -1.75045   1.06406   1.63112  -1.24698  -1.49128   1.41421   1.33269  -1.56366  -1.15734   1.69345   0.96744  -1.80194  -0.76537   1.88777   0.55367  -1.94986  -0.33501   1.98742   0.11214  -2.00000     ... ... ..... ....... ......... ........... .............
E30 *    2.00000  -0.22393  -1.94986   0.66056   1.80194  -1.06406  -1.56366   1.41421   1.24698  -1.69345  -0.86777   1.88777   0.44504  -1.98742   0.00000   1.98742  -0.44504  -1.88777   0.86777   1.69345  -1.24698  -1.41421   1.56366   1.06406  -1.80194  -0.66056   1.94986   0.22393  -2.00000   0.22393   1.94986  -0.66056  -1.80194   1.06406   1.56366  -1.41421  -1.24698   1.69345   0.86777  -1.88777  -0.44504   1.98742   0.00000  -1.98742   0.44504   1.88777  -0.86777  -1.69345   1.24698   1.41421  -1.56366  -1.06406   1.80194   0.66056  -1.94986  -0.22393   2.00000     ... ... ..... ....... ......... ........... .............
E31 *    2.00000  -0.33501  -1.88777   0.96744   1.56366  -1.49128  -1.06406   1.84776   0.44504  -1.99685   0.22393   1.92183  -0.86777  -1.63112   1.41421   1.15734  -1.80194  -0.55367   1.98742  -0.11214  -1.94986   0.76537   1.69345  -1.33269  -1.24698   1.75045   0.66056  -1.97174   0.00000   1.97174  -0.66056  -1.75045   1.24698   1.33269  -1.69345  -0.76537   1.94986   0.11214  -1.98742   0.55367   1.80194  -1.15734  -1.41421   1.63112   0.86777  -1.92183  -0.22393   1.99685  -0.44504  -1.84776   1.06406   1.49128  -1.56366  -0.96744   1.88777   0.33501  -2.00000     ... ... ..... ....... ......... ........... .............
E32 *    2.00000  -0.44504  -1.80194   1.24698   1.24698  -1.80194  -0.44504   2.00000  -0.44504  -1.80194   1.24698   1.24698  -1.80194  -0.44504   2.00000  -0.44504  -1.80194   1.24698   1.24698  -1.80194  -0.44504   2.00000  -0.44504  -1.80194   1.24698   1.24698  -1.80194  -0.44504   2.00000  -0.44504  -1.80194   1.24698   1.24698  -1.80194  -0.44504   2.00000  -0.44504  -1.80194   1.24698   1.24698  -1.80194  -0.44504   2.00000  -0.44504  -1.80194   1.24698   1.24698  -1.80194  -0.44504   2.00000  -0.44504  -1.80194   1.24698   1.24698  -1.80194  -0.44504   2.00000     ... ... ..... ....... ......... ........... .............
E33 *    2.00000  -0.55367  -1.69345   1.49128   0.86777  -1.97174   0.22393   1.84776  -1.24698  -1.15734   1.88777   0.11214  -1.94986   0.96744   1.41421  -1.75045  -0.44504   1.99685  -0.66056  -1.63112   1.56366   0.76537  -1.98742   0.33501   1.80194  -1.33269  -1.06406   1.92183   0.00000  -1.92183   1.06406   1.33269  -1.80194  -0.33501   1.98742  -0.76537  -1.56366   1.63112   0.66056  -1.99685   0.44504   1.75045  -1.41421  -0.96744   1.94986  -0.11214  -1.88777   1.15734   1.24698  -1.84776  -0.22393   1.97174  -0.86777  -1.49128   1.69345   0.55367  -2.00000     ... ... ..... ....... ......... ........... .............
E34 *    2.00000  -0.66056  -1.56366   1.69345   0.44504  -1.98742   0.86777   1.41421  -1.80194  -0.22393   1.94986  -1.06406  -1.24698   1.88777   0.00000  -1.88777   1.24698   1.06406  -1.94986   0.22393   1.80194  -1.41421  -0.86777   1.98742  -0.44504  -1.69345   1.56366   0.66056  -2.00000   0.66056   1.56366  -1.69345  -0.44504   1.98742  -0.86777  -1.41421   1.80194   0.22393  -1.94986   1.06406   1.24698  -1.88777   0.00000   1.88777  -1.24698  -1.06406   1.94986  -0.22393  -1.80194   1.41421   0.86777  -1.98742   0.44504   1.69345  -1.56366  -0.66056   2.00000     ... ... ..... ....... ......... ........... .............
E35 *    2.00000  -0.76537  -1.41421   1.84776   0.00000  -1.84776   1.41421   0.76537  -2.00000   0.76537   1.41421  -1.84776   0.00000   1.84776  -1.41421  -0.76537   2.00000  -0.76537  -1.41421   1.84776   0.00000  -1.84776   1.41421   0.76537  -2.00000   0.76537   1.41421  -1.84776   0.00000   1.84776  -1.41421  -0.76537   2.00000  -0.76537  -1.41421   1.84776   0.00000  -1.84776   1.41421   0.76537  -2.00000   0.76537   1.41421  -1.84776   0.00000   1.84776  -1.41421  -0.76537   2.00000  -0.76537  -1.41421   1.84776   0.00000  -1.84776   1.41421   0.76537  -2.00000     ... ... ..... ....... ......... ........... .............
E36 *    2.00000  -0.86777  -1.24698   1.94986  -0.44504  -1.56366   1.80194   0.00000  -1.80194   1.56366   0.44504  -1.94986   1.24698   0.86777  -2.00000   0.86777   1.24698  -1.94986   0.44504   1.56366  -1.80194   0.00000   1.80194  -1.56366  -0.44504   1.94986  -1.24698  -0.86777   2.00000  -0.86777  -1.24698   1.94986  -0.44504  -1.56366   1.80194   0.00000  -1.80194   1.56366   0.44504  -1.94986   1.24698   0.86777  -2.00000   0.86777   1.24698  -1.94986   0.44504   1.56366  -1.80194   0.00000   1.80194  -1.56366  -0.44504   1.94986  -1.24698  -0.86777   2.00000     ... ... ..... ....... ......... ........... .............
E37 *    2.00000  -0.96744  -1.06406   1.99685  -0.86777  -1.15734   1.98742  -0.76537  -1.24698   1.97174  -0.66056  -1.33269   1.94986  -0.55367  -1.41421   1.92183  -0.44504  -1.49128   1.88777  -0.33501  -1.56366   1.84776  -0.22393  -1.63112   1.80194  -0.11214  -1.69345   1.75045   0.00000  -1.75045   1.69345   0.11214  -1.80194   1.63112   0.22393  -1.84776   1.56366   0.33501  -1.88777   1.49128   0.44504  -1.92183   1.41421   0.55367  -1.94986   1.33269   0.66056  -1.97174   1.24698   0.76537  -1.98742   1.15734   0.86777  -1.99685   1.06406   0.96744  -2.00000     ... ... ..... ....... ......... ........... .............
E38 *    2.00000  -1.06406  -0.86777   1.98742  -1.24698  -0.66056   1.94986  -1.41421  -0.44504   1.88777  -1.56366  -0.22393   1.80194  -1.69345   0.00000   1.69345  -1.80194   0.22393   1.56366  -1.88777   0.44504   1.41421  -1.94986   0.66056   1.24698  -1.98742   0.86777   1.06406  -2.00000   1.06406   0.86777  -1.98742   1.24698   0.66056  -1.94986   1.41421   0.44504  -1.88777   1.56366   0.22393  -1.80194   1.69345   0.00000  -1.69345   1.80194  -0.22393  -1.56366   1.88777  -0.44504  -1.41421   1.94986  -0.66056  -1.24698   1.98742  -0.86777  -1.06406   2.00000     ... ... ..... ....... ......... ........... .............
E39 *    2.00000  -1.15734  -0.66056   1.92183  -1.56366  -0.11214   1.69345  -1.84776   0.44504   1.33269  -1.98742   0.96744   0.86777  -1.97174   1.41421   0.33501  -1.80194   1.75045  -0.22393  -1.49128   1.94986  -0.76537  -1.06406   1.99685  -1.24698  -0.55367   1.88777  -1.63112   0.00000   1.63112  -1.88777   0.55367   1.24698  -1.99685   1.06406   0.76537  -1.94986   1.49128   0.22393  -1.75045   1.80194  -0.33501  -1.41421   1.97174  -0.86777  -0.96744   1.98742  -1.33269  -0.44504   1.84776  -1.69345   0.11214   1.56366  -1.92183   0.66056   1.15734  -2.00000     ... ... ..... ....... ......... ........... .............
E40 *    2.00000  -1.24698  -0.44504   1.80194  -1.80194   0.44504   1.24698  -2.00000   1.24698   0.44504  -1.80194   1.80194  -0.44504  -1.24698   2.00000  -1.24698  -0.44504   1.80194  -1.80194   0.44504   1.24698  -2.00000   1.24698   0.44504  -1.80194   1.80194  -0.44504  -1.24698   2.00000  -1.24698  -0.44504   1.80194  -1.80194   0.44504   1.24698  -2.00000   1.24698   0.44504  -1.80194   1.80194  -0.44504  -1.24698   2.00000  -1.24698  -0.44504   1.80194  -1.80194   0.44504   1.24698  -2.00000   1.24698   0.44504  -1.80194   1.80194  -0.44504  -1.24698   2.00000     ... ... ..... ....... ......... ........... .............
E41 *    2.00000  -1.33269  -0.22393   1.63112  -1.94986   0.96744   0.66056  -1.84776   1.80194  -0.55367  -1.06406   1.97174  -1.56366   0.11214   1.41421  -1.99685   1.24698   0.33501  -1.69345   1.92183  -0.86777  -0.76537   1.88777  -1.75045   0.44504   1.15734  -1.98742   1.49128   0.00000  -1.49128   1.98742  -1.15734  -0.44504   1.75045  -1.88777   0.76537   0.86777  -1.92183   1.69345  -0.33501  -1.24698   1.99685  -1.41421  -0.11214   1.56366  -1.97174   1.06406   0.55367  -1.80194   1.84776  -0.66056  -0.96744   1.94986  -1.63112   0.22393   1.33269  -2.00000     ... ... ..... ....... ......... ........... .............
E42 *    2.00000  -1.41421   0.00000   1.41421  -2.00000   1.41421   0.00000  -1.41421   2.00000  -1.41421   0.00000   1.41421  -2.00000   1.41421   0.00000  -1.41421   2.00000  -1.41421   0.00000   1.41421  -2.00000   1.41421   0.00000  -1.41421   2.00000  -1.41421   0.00000   1.41421  -2.00000   1.41421   0.00000  -1.41421   2.00000  -1.41421   0.00000   1.41421  -2.00000   1.41421   0.00000  -1.41421   2.00000  -1.41421   0.00000   1.41421  -2.00000   1.41421   0.00000  -1.41421   2.00000  -1.41421   0.00000   1.41421  -2.00000   1.41421   0.00000  -1.41421   2.00000     ... ... ..... ....... ......... ........... .............
E43 *    2.00000  -1.49128   0.22393   1.15734  -1.94986   1.75045  -0.66056  -0.76537   1.80194  -1.92183   1.06406   0.33501  -1.56366   1.99685  -1.41421   0.11214   1.24698  -1.97174   1.69345  -0.55367  -0.86777   1.84776  -1.88777   0.96744   0.44504  -1.63112   1.98742  -1.33269   0.00000   1.33269  -1.98742   1.63112  -0.44504  -0.96744   1.88777  -1.84776   0.86777   0.55367  -1.69345   1.97174  -1.24698  -0.11214   1.41421  -1.99685   1.56366  -0.33501  -1.06406   1.92183  -1.80194   0.76537   0.66056  -1.75045   1.94986  -1.15734  -0.22393   1.49128  -2.00000     ... ... ..... ....... ......... ........... .............
E44 *    2.00000  -1.56366   0.44504   0.86777  -1.80194   1.94986  -1.24698   0.00000   1.24698  -1.94986   1.80194  -0.86777  -0.44504   1.56366  -2.00000   1.56366  -0.44504  -0.86777   1.80194  -1.94986   1.24698   0.00000  -1.24698   1.94986  -1.80194   0.86777   0.44504  -1.56366   2.00000  -1.56366   0.44504   0.86777  -1.80194   1.94986  -1.24698   0.00000   1.24698  -1.94986   1.80194  -0.86777  -0.44504   1.56366  -2.00000   1.56366  -0.44504  -0.86777   1.80194  -1.94986   1.24698   0.00000  -1.24698   1.94986  -1.80194   0.86777   0.44504  -1.56366   2.00000     ... ... ..... ....... ......... ........... .............
E45 *    2.00000  -1.63112   0.66056   0.55367  -1.56366   1.99685  -1.69345   0.76537   0.44504  -1.49128   1.98742  -1.75045   0.86777   0.33501  -1.41421   1.97174  -1.80194   0.96744   0.22393  -1.33269   1.94986  -1.84776   1.06406   0.11214  -1.24698   1.92183  -1.88777   1.15734   0.00000  -1.15734   1.88777  -1.92183   1.24698  -0.11214  -1.06406   1.84776  -1.94986   1.33269  -0.22393  -0.96744   1.80194  -1.97174   1.41421  -0.33501  -0.86777   1.75045  -1.98742   1.49128  -0.44504  -0.76537   1.69345  -1.99685   1.56366  -0.55367  -0.66056   1.63112  -2.00000     ... ... ..... ....... ......... ........... .............
E46 *    2.00000  -1.69345   0.86777   0.22393  -1.24698   1.88777  -1.94986   1.41421  -0.44504  -0.66056   1.56366  -1.98742   1.80194  -1.06406   0.00000   1.06406  -1.80194   1.98742  -1.56366   0.66056   0.44504  -1.41421   1.94986  -1.88777   1.24698  -0.22393  -0.86777   1.69345  -2.00000   1.69345  -0.86777  -0.22393   1.24698  -1.88777   1.94986  -1.41421   0.44504   0.66056  -1.56366   1.98742  -1.80194   1.06406   0.00000  -1.06406   1.80194  -1.98742   1.56366  -0.66056  -0.44504   1.41421  -1.94986   1.88777  -1.24698   0.22393   0.86777  -1.69345   2.00000     ... ... ..... ....... ......... ........... .............
E47 *    2.00000  -1.75045   1.06406  -0.11214  -0.86777   1.63112  -1.98742   1.84776  -1.24698   0.33501   0.66056  -1.49128   1.94986  -1.92183   1.41421  -0.55367  -0.44504   1.33269  -1.88777   1.97174  -1.56366   0.76537   0.22393  -1.15734   1.80194  -1.99685   1.69345  -0.96744   0.00000   0.96744  -1.69345   1.99685  -1.80194   1.15734  -0.22393  -0.76537   1.56366  -1.97174   1.88777  -1.33269   0.44504   0.55367  -1.41421   1.92183  -1.94986   1.49128  -0.66056  -0.33501   1.24698  -1.84776   1.98742  -1.63112   0.86777   0.11214  -1.06406   1.75045  -2.00000     ... ... ..... ....... ......... ........... .............
E48 *    2.00000  -1.80194   1.24698  -0.44504  -0.44504   1.24698  -1.80194   2.00000  -1.80194   1.24698  -0.44504  -0.44504   1.24698  -1.80194   2.00000  -1.80194   1.24698  -0.44504  -0.44504   1.24698  -1.80194   2.00000  -1.80194   1.24698  -0.44504  -0.44504   1.24698  -1.80194   2.00000  -1.80194   1.24698  -0.44504  -0.44504   1.24698  -1.80194   2.00000  -1.80194   1.24698  -0.44504  -0.44504   1.24698  -1.80194   2.00000  -1.80194   1.24698  -0.44504  -0.44504   1.24698  -1.80194   2.00000  -1.80194   1.24698  -0.44504  -0.44504   1.24698  -1.80194   2.00000     ... ... ..... ....... ......... ........... .............
E49 *    2.00000  -1.84776   1.41421  -0.76537   0.00000   0.76537  -1.41421   1.84776  -2.00000   1.84776  -1.41421   0.76537   0.00000  -0.76537   1.41421  -1.84776   2.00000  -1.84776   1.41421  -0.76537   0.00000   0.76537  -1.41421   1.84776  -2.00000   1.84776  -1.41421   0.76537   0.00000  -0.76537   1.41421  -1.84776   2.00000  -1.84776   1.41421  -0.76537   0.00000   0.76537  -1.41421   1.84776  -2.00000   1.84776  -1.41421   0.76537   0.00000  -0.76537   1.41421  -1.84776   2.00000  -1.84776   1.41421  -0.76537   0.00000   0.76537  -1.41421   1.84776  -2.00000     ... ... ..... ....... ......... ........... .............
E50 *    2.00000  -1.88777   1.56366  -1.06406   0.44504   0.22393  -0.86777   1.41421  -1.80194   1.98742  -1.94986   1.69345  -1.24698   0.66056   0.00000  -0.66056   1.24698  -1.69345   1.94986  -1.98742   1.80194  -1.41421   0.86777  -0.22393  -0.44504   1.06406  -1.56366   1.88777  -2.00000   1.88777  -1.56366   1.06406  -0.44504  -0.22393   0.86777  -1.41421   1.80194  -1.98742   1.94986  -1.69345   1.24698  -0.66056   0.00000   0.66056  -1.24698   1.69345  -1.94986   1.98742  -1.80194   1.41421  -0.86777   0.22393   0.44504  -1.06406   1.56366  -1.88777   2.00000     ... ... ..... ....... ......... ........... .............
E51 *    2.00000  -1.92183   1.69345  -1.33269   0.86777  -0.33501  -0.22393   0.76537  -1.24698   1.63112  -1.88777   1.99685  -1.94986   1.75045  -1.41421   0.96744  -0.44504  -0.11214   0.66056  -1.15734   1.56366  -1.84776   1.98742  -1.97174   1.80194  -1.49128   1.06406  -0.55367   0.00000   0.55367  -1.06406   1.49128  -1.80194   1.97174  -1.98742   1.84776  -1.56366   1.15734  -0.66056   0.11214   0.44504  -0.96744   1.41421  -1.75045   1.94986  -1.99685   1.88777  -1.63112   1.24698  -0.76537   0.22393   0.33501  -0.86777   1.33269  -1.69345   1.92183  -2.00000     ... ... ..... ....... ......... ........... ..TT.........
E52 *    2.00000  -1.94986   1.80194  -1.56366   1.24698  -0.86777   0.44504   0.00000  -0.44504   0.86777  -1.24698   1.56366  -1.80194   1.94986  -2.00000   1.94986  -1.80194   1.56366  -1.24698   0.86777  -0.44504   0.00000   0.44504  -0.86777   1.24698  -1.56366   1.80194  -1.94986   2.00000  -1.94986   1.80194  -1.56366   1.24698  -0.86777   0.44504   0.00000  -0.44504   0.86777  -1.24698   1.56366  -1.80194   1.94986  -2.00000   1.94986  -1.80194   1.56366  -1.24698   0.86777  -0.44504   0.00000   0.44504  -0.86777   1.24698  -1.56366   1.80194  -1.94986   2.00000     ... ... ..... ....... ......... ..TT....... .............
E53 *    2.00000  -1.97174   1.88777  -1.75045   1.56366  -1.33269   1.06406  -0.76537   0.44504  -0.11214  -0.22393   0.55367  -0.86777   1.15734  -1.41421   1.63112  -1.80194   1.92183  -1.98742   1.99685  -1.94986   1.84776  -1.69345   1.49128  -1.24698   0.96744  -0.66056   0.33501   0.00000  -0.33501   0.66056  -0.96744   1.24698  -1.49128   1.69345  -1.84776   1.94986  -1.99685   1.98742  -1.92183   1.80194  -1.63112   1.41421  -1.15734   0.86777  -0.55367   0.22393   0.11214  -0.44504   0.76537  -1.06406   1.33269  -1.56366   1.75045  -1.88777   1.97174  -2.00000     ... ... ..... ....... ..TT..... ........... ......TT.....
E54 *    2.00000  -1.98742   1.94986  -1.88777   1.80194  -1.69345   1.56366  -1.41421   1.24698  -1.06406   0.86777  -0.66056   0.44504  -0.22393   0.00000   0.22393  -0.44504   0.66056  -0.86777   1.06406  -1.24698   1.41421  -1.56366   1.69345  -1.80194   1.88777  -1.94986   1.98742  -2.00000   1.98742  -1.94986   1.88777  -1.80194   1.69345  -1.56366   1.41421  -1.24698   1.06406  -0.86777   0.66056  -0.44504   0.22393   0.00000  -0.22393   0.44504  -0.66056   0.86777  -1.06406   1.24698  -1.41421   1.56366  -1.69345   1.80194  -1.88777   1.94986  -1.98742   2.00000     ... ... ..... ..TT... ......... ......TT... .............
E55 *    2.00000  -1.99685   1.98742  -1.97174   1.94986  -1.92183   1.88777  -1.84776   1.80194  -1.75045   1.69345  -1.63112   1.56366  -1.49128   1.41421  -1.33269   1.24698  -1.15734   1.06406  -0.96744   0.86777  -0.76537   0.66056  -0.55367   0.44504  -0.33501   0.22393  -0.11214   0.00000   0.11214  -0.22393   0.33501  -0.44504   0.55367  -0.66056   0.76537  -0.86777   0.96744  -1.06406   1.15734  -1.24698   1.33269  -1.41421   1.49128  -1.56366   1.63112  -1.69345   1.75045  -1.80194   1.84776  -1.88777   1.92183  -1.94986   1.97174  -1.98742   1.99685  -2.00000     TT. ... ..TT. ....... ......TT. ........... ..........TT.

 Irrational character values:  1.996853630036 = 2*cos(2*π/112) = 2*cos(π/56)
                               1.987424419786 = 2*cos(4*π/112) = 2*cos(π/28)
                               1.971742037036 = 2*cos(6*π/112) = 2*cos(3*π/56)
                               1.949855824364 = 2*cos(8*π/112) = 2*cos(π/14)
                               1.921834643890 = 2*cos(10*π/112) = 2*cos(5*π/56)
                               1.887766660617 = 2*cos(12*π/112) = 2*cos(3*π/28)
                               1.847759065023 = 2*cos(14*π/112) = 2*cos(π/8) = √2+√2
                               1.801937735805 = 2*cos(16*π/112) = 2*cos(π/7)
                               1.750446843817 = 2*cos(18*π/112) = 2*cos(9*π/56)
                               1.693448398457 = 2*cos(20*π/112) = 2*cos(5*π/28)
                               1.631121737918 = 2*cos(22*π/112) = 2*cos(11*π/56)
                               1.563662964936 = 2*cos(24*π/112) = 2*cos(3*π/14)
                               1.491284329766 = 2*cos(26*π/112) = 2*cos(13*π/56)
                               1.414213562373 = 2*cos(28*π/112) = 2*cos(π/4) = √2
                               1.332693155904 = 2*cos(30*π/112) = 2*cos(15*π/56)
                               1.246979603717 = 2*cos(32*π/112) = 2*cos(2*π/7)
                               1.157342592360 = 2*cos(34*π/112) = 2*cos(17*π/56)
                               1.064064153031 = 2*cos(36*π/112) = 2*cos(9*π/28)
                               0.967437774210 = 2*cos(38*π/112) = 2*cos(19*π/56)
                               0.867767478235 = 2*cos(40*π/112) = 2*cos(5*π/14)
                               0.765366864730 = 2*cos(42*π/112) = 2*cos(3*π/8) = √2-√2
                               0.660558123910 = 2*cos(44*π/112) = 2*cos(11*π/28)
                               0.553671022850 = 2*cos(46*π/112) = 2*cos(23*π/56)
                               0.445041867912 = 2*cos(48*π/112) = 2*cos(3*π/7)
                               0.335012446609 = 2*cos(50*π/112) = 2*cos(25*π/56)
                               0.223928952206 = 2*cos(52*π/112) = 2*cos(13*π/28)
                               0.112140894474 = 2*cos(54*π/112) = 2*cos(27*π/56)



 Symmetry of Rotations and Cartesian products

A    R+d+g+i  Rz, z2, z4, z6 
B    p+f+h    z, z3, z5 
E1   p+f+h    {x, y}, {xz2, yz2}, {xz4, yz4} 
E2   d+g+i    {x2y2, xy}, {z2(x2y2), xyz2}, {z4(x2y2), xyz4} 
E3   f+h      {x(x2–3y2), y(3x2y2)}, {xz2(x2–3y2), yz2(3x2y2)} 
E4   g+i      {(x2y2)2–2x2y2, xy(x2y2)}, {z2((x2y2)2–4x2y2), xyz2(x2y2)} 
E5   h        {x(x2–(5+2√5)y2)(x2–(5–2√5)y2), y(y2–(5+2√5)x2)(y2–(5–2√5)x2)(y2–(5–2√5)x2)} 
E6   i        {x2(x2–3y2)2y2(3x2y2)2, xy(x2–3y2)(3x2y2)} 
E51  i        {xz(x2–(5+2√5)y2)(x2–(5–2√5)y2), yz(y2–(5+2√5)x2)(y2–(5–2√5)x2)(y2–(5–2√5)x2)} 
E52  h        {z((x2y2)2–4x2y2), xyz(x2y2)} 
E53  g+i      {xz(x2–3y2), yz(3x2y2)}, {xz3(x2–3y2), yz3(3x2y2)} 
E54  f+h      {z(x2y2), xyz}, {z3(x2y2), xyz3} 
E55  R+d+g+i  {Rx, Ry}, {xz, yz}, {xz3, yz3}, {xz5, yz5} 

 Notes:

    α  The order of the S112 point group is 112, and the order of the principal axis is 112.

    β  The S112 point group could also be named C112i, because the S112 axis is identical to a roto-inversion axis of order 112.

    γ  This is an Abelian point group (the commutative law holds between all symmetry operations).
       The S112 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.

    δ  Because the group is Abelian and the maximum order of rotation is >2, some irreducible representations have complex characters.
       These 110 cases have been combined into 55 two-dimensional representations that are no longer irreducible but have real-valued characters.
       Accordingly, 55 pairs of left and right rotations have been combined into one two-membered pseudo-class each.

    ε  The 55 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.

    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 less than half the order of the principal axis (and often much less).
       For this group, however, some of the characters cannot be expressed by a closed algebraic form using real numbers only.

    η  The regular heptagon is the lowest regular polygon that cannot be constructed by compass and ruler alone, which has mystified
       mathematicians since antiquity. The reason for the inconstructibility of the heptagon is that cos(2π/7) has an algebraic degree of 3 and
       thus can not be represented by square roots and integer numbers. An algebraic representation becomes possible if cubic roots and complex numbers
       are used: 2*cos(2π/7) = (328+i*84*√3 + 328-i*84*√3 - 2)/6. This complex expression for a real value is hardly useful.
       Therefore, regular polygons of order 14,21,28,35,42 etc. are also inconstructible, and their cosines have no representation in real radicals.

This Character Table for the S112 point group was created by Gernot Katzer.

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