D69 E 2 C69 2 C69^2 2 C23 2 C69^4 2 C69^5 2 C23^2 2 C69^7 2 C69^8 2 C23^3 2 C69^10 2 C69^11 2 C23^4 2 C69^13 2 C69^14 2 C23^5 2 C69^16 2 C69^17 2 C23^6 2 C69^19 2 C69^20 2 C23^7 2 C69^22 2 C3 2 C23^8 2 C69^25 2 C69^26 2 C23^9 2 C69^28 2 C69^29 2 C23^10 2 C69^31 2 C69^32 2 C23^11 2 C69^34 69 C2' <R> <p> <—d—> <——f——> <———g———> <————h————> <—————i—————> A1 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 ... ... ....T ....... ........T ........... ............T A2 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 -1.00000 ..T ..T ..... ......T ......... ..........T ............. E1 2.00000 1.99171 1.96692 1.92583 1.86879 1.79626 1.70884 1.60726 1.49237 1.36511 1.22653 1.07780 0.92013 0.75484 0.58329 0.40691 0.22716 0.04553 -0.13648 -0.31737 -0.49562 -0.66976 -0.83835 -1.00000 -1.15336 -1.29716 -1.43022 -1.55142 -1.65977 -1.75437 -1.83442 -1.89928 -1.94840 -1.98137 -1.99793 0.00000 TT. TT. ..TT. ....TT. ......TT. ........TT. ..........TT. E2 2.00000 1.96692 1.86879 1.70884 1.49237 1.22653 0.92013 0.58329 0.22716 -0.13648 -0.49562 -0.83835 -1.15336 -1.43022 -1.65977 -1.83442 -1.94840 -1.99793 -1.98137 -1.89928 -1.75437 -1.55142 -1.29716 -1.00000 -0.66976 -0.31737 0.04553 0.40691 0.75484 1.07780 1.36511 1.60726 1.79626 1.92583 1.99171 0.00000 ... ... TT... ..TT... ....TT... ......TT... ........TT... E3 2.00000 1.92583 1.70884 1.36511 0.92013 0.40691 -0.13648 -0.66976 -1.15336 -1.55142 -1.83442 -1.98137 -1.98137 -1.83442 -1.55142 -1.15336 -0.66976 -0.13648 0.40691 0.92013 1.36511 1.70884 1.92583 2.00000 1.92583 1.70884 1.36511 0.92013 0.40691 -0.13648 -0.66976 -1.15336 -1.55142 -1.83442 -1.98137 0.00000 ... ... ..... TT..... ..TT..... ....TT..... ......TT..... E4 2.00000 1.86879 1.49237 0.92013 0.22716 -0.49562 -1.15336 -1.65977 -1.94840 -1.98137 -1.75437 -1.29716 -0.66976 0.04553 0.75484 1.36511 1.79626 1.99171 1.92583 1.60726 1.07780 0.40691 -0.31737 -1.00000 -1.55142 -1.89928 -1.99793 -1.83442 -1.43022 -0.83835 -0.13648 0.58329 1.22653 1.70884 1.96692 0.00000 ... ... ..... ....... TT....... ..TT....... ....TT....... E5 2.00000 1.79626 1.22653 0.40691 -0.49562 -1.29716 -1.83442 -1.99793 -1.75437 -1.15336 -0.31737 0.58329 1.36511 1.86879 1.99171 1.70884 1.07780 0.22716 -0.66976 -1.43022 -1.89928 -1.98137 -1.65977 -1.00000 -0.13648 0.75484 1.49237 1.92583 1.96692 1.60726 0.92013 0.04553 -0.83835 -1.55142 -1.94840 0.00000 ... ... ..... ....... ......... TT......... ..TT......... E6 2.00000 1.70884 0.92013 -0.13648 -1.15336 -1.83442 -1.98137 -1.55142 -0.66976 0.40691 1.36511 1.92583 1.92583 1.36511 0.40691 -0.66976 -1.55142 -1.98137 -1.83442 -1.15336 -0.13648 0.92013 1.70884 2.00000 1.70884 0.92013 -0.13648 -1.15336 -1.83442 -1.98137 -1.55142 -0.66976 0.40691 1.36511 1.92583 0.00000 ... ... ..... ....... ......... ........... TT........... E7 2.00000 1.60726 0.58329 -0.66976 -1.65977 -1.99793 -1.55142 -0.49562 0.75484 1.70884 1.99171 1.49237 0.40691 -0.83835 -1.75437 -1.98137 -1.43022 -0.31737 0.92013 1.79626 1.96692 1.36511 0.22716 -1.00000 -1.83442 -1.94840 -1.29716 -0.13648 1.07780 1.86879 1.92583 1.22653 0.04553 -1.15336 -1.89928 0.00000 ... ... ..... ....... ......... ........... ............. E8 2.00000 1.49237 0.22716 -1.15336 -1.94840 -1.75437 -0.66976 0.75484 1.79626 1.92583 1.07780 -0.31737 -1.55142 -1.99793 -1.43022 -0.13648 1.22653 1.96692 1.70884 0.58329 -0.83835 -1.83442 -1.89928 -1.00000 0.40691 1.60726 1.99171 1.36511 0.04553 -1.29716 -1.98137 -1.65977 -0.49562 0.92013 1.86879 0.00000 ... ... ..... ....... ......... ........... ............. E9 2.00000 1.36511 -0.13648 -1.55142 -1.98137 -1.15336 0.40691 1.70884 1.92583 0.92013 -0.66976 -1.83442 -1.83442 -0.66976 0.92013 1.92583 1.70884 0.40691 -1.15336 -1.98137 -1.55142 -0.13648 1.36511 2.00000 1.36511 -0.13648 -1.55142 -1.98137 -1.15336 0.40691 1.70884 1.92583 0.92013 -0.66976 -1.83442 0.00000 ... ... ..... ....... ......... ........... ............. E10 2.00000 1.22653 -0.49562 -1.83442 -1.75437 -0.31737 1.36511 1.99171 1.07780 -0.66976 -1.89928 -1.65977 -0.13648 1.49237 1.96692 0.92013 -0.83835 -1.94840 -1.55142 0.04553 1.60726 1.92583 0.75484 -1.00000 -1.98137 -1.43022 0.22716 1.70884 1.86879 0.58329 -1.15336 -1.99793 -1.29716 0.40691 1.79626 0.00000 ... ... ..... ....... ......... ........... ............. E11 2.00000 1.07780 -0.83835 -1.98137 -1.29716 0.58329 1.92583 1.49237 -0.31737 -1.83442 -1.65977 0.04553 1.70884 1.79626 0.22716 -1.55142 -1.89928 -0.49562 1.36511 1.96692 0.75484 -1.15336 -1.99793 -1.00000 0.92013 1.99171 1.22653 -0.66976 -1.94840 -1.43022 0.40691 1.86879 1.60726 -0.13648 -1.75437 0.00000 ... ... ..... ....... ......... ........... ............. E12 2.00000 0.92013 -1.15336 -1.98137 -0.66976 1.36511 1.92583 0.40691 -1.55142 -1.83442 -0.13648 1.70884 1.70884 -0.13648 -1.83442 -1.55142 0.40691 1.92583 1.36511 -0.66976 -1.98137 -1.15336 0.92013 2.00000 0.92013 -1.15336 -1.98137 -0.66976 1.36511 1.92583 0.40691 -1.55142 -1.83442 -0.13648 1.70884 0.00000 ... ... ..... ....... ......... ........... ............. E13 2.00000 0.75484 -1.43022 -1.83442 0.04553 1.86879 1.36511 -0.83835 -1.99793 -0.66976 1.49237 1.79626 -0.13648 -1.89928 -1.29716 0.92013 1.99171 0.58329 -1.55142 -1.75437 0.22716 1.92583 1.22653 -1.00000 -1.98137 -0.49562 1.60726 1.70884 -0.31737 -1.94840 -1.15336 1.07780 1.96692 0.40691 -1.65977 0.00000 ... ... ..... ....... ......... ........... ............. E14 2.00000 0.58329 -1.65977 -1.55142 0.75484 1.99171 0.40691 -1.75437 -1.43022 0.92013 1.96692 0.22716 -1.83442 -1.29716 1.07780 1.92583 0.04553 -1.89928 -1.15336 1.22653 1.86879 -0.13648 -1.94840 -1.00000 1.36511 1.79626 -0.31737 -1.98137 -0.83835 1.49237 1.70884 -0.49562 -1.99793 -0.66976 1.60726 0.00000 ... ... ..... ....... ......... ........... ............. E15 2.00000 0.40691 -1.83442 -1.15336 1.36511 1.70884 -0.66976 -1.98137 -0.13648 1.92583 0.92013 -1.55142 -1.55142 0.92013 1.92583 -0.13648 -1.98137 -0.66976 1.70884 1.36511 -1.15336 -1.83442 0.40691 2.00000 0.40691 -1.83442 -1.15336 1.36511 1.70884 -0.66976 -1.98137 -0.13648 1.92583 0.92013 -1.55142 0.00000 ... ... ..... ....... ......... ........... ............. E16 2.00000 0.22716 -1.94840 -0.66976 1.79626 1.07780 -1.55142 -1.43022 1.22653 1.70884 -0.83835 -1.89928 0.40691 1.99171 0.04553 -1.98137 -0.49562 1.86879 0.92013 -1.65977 -1.29716 1.36511 1.60726 -1.00000 -1.83442 0.58329 1.96692 -0.13648 -1.99793 -0.31737 1.92583 0.75484 -1.75437 -1.15336 1.49237 0.00000 ... ... ..... ....... ......... ........... ............. E17 2.00000 0.04553 -1.99793 -0.13648 1.99171 0.22716 -1.98137 -0.31737 1.96692 0.40691 -1.94840 -0.49562 1.92583 0.58329 -1.89928 -0.66976 1.86879 0.75484 -1.83442 -0.83835 1.79626 0.92013 -1.75437 -1.00000 1.70884 1.07780 -1.65977 -1.15336 1.60726 1.22653 -1.55142 -1.29716 1.49237 1.36511 -1.43022 0.00000 ... ... ..... ....... ......... ........... ............. E18 2.00000 -0.13648 -1.98137 0.40691 1.92583 -0.66976 -1.83442 0.92013 1.70884 -1.15336 -1.55142 1.36511 1.36511 -1.55142 -1.15336 1.70884 0.92013 -1.83442 -0.66976 1.92583 0.40691 -1.98137 -0.13648 2.00000 -0.13648 -1.98137 0.40691 1.92583 -0.66976 -1.83442 0.92013 1.70884 -1.15336 -1.55142 1.36511 0.00000 ... ... ..... ....... ......... ........... ............. E19 2.00000 -0.31737 -1.89928 0.92013 1.60726 -1.43022 -1.15336 1.79626 0.58329 -1.98137 0.04553 1.96692 -0.66976 -1.75437 1.22653 1.36511 -1.65977 -0.83835 1.92583 0.22716 -1.99793 0.40691 1.86879 -1.00000 -1.55142 1.49237 1.07780 -1.83442 -0.49562 1.99171 -0.13648 -1.94840 0.75484 1.70884 -1.29716 0.00000 ... ... ..... ....... ......... ........... ............. E20 2.00000 -0.49562 -1.75437 1.36511 1.07780 -1.89928 -0.13648 1.96692 -0.83835 -1.55142 1.60726 0.75484 -1.98137 0.22716 1.86879 -1.15336 -1.29716 1.79626 0.40691 -1.99793 0.58329 1.70884 -1.43022 -1.00000 1.92583 0.04553 -1.94840 0.92013 1.49237 -1.65977 -0.66976 1.99171 -0.31737 -1.83442 1.22653 0.00000 ... ... ..... ....... ......... ........... ............. E21 2.00000 -0.66976 -1.55142 1.70884 0.40691 -1.98137 0.92013 1.36511 -1.83442 -0.13648 1.92583 -1.15336 -1.15336 1.92583 -0.13648 -1.83442 1.36511 0.92013 -1.98137 0.40691 1.70884 -1.55142 -0.66976 2.00000 -0.66976 -1.55142 1.70884 0.40691 -1.98137 0.92013 1.36511 -1.83442 -0.13648 1.92583 -1.15336 0.00000 ... ... ..... ....... ......... ........... ............. E22 2.00000 -0.83835 -1.29716 1.92583 -0.31737 -1.65977 1.70884 0.22716 -1.89928 1.36511 0.75484 -1.99793 0.92013 1.22653 -1.94840 0.40691 1.60726 -1.75437 -0.13648 1.86879 -1.43022 -0.66976 1.99171 -1.00000 -1.15336 1.96692 -0.49562 -1.55142 1.79626 0.04553 -1.83442 1.49237 0.58329 -1.98137 1.07780 0.00000 ... ... ..... ....... ......... ........... ............. E23 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 -1.00000 2.00000 -1.00000 -1.00000 2.00000 -1.00000 -1.00000 2.00000 -1.00000 0.00000 ... ... ..... ....... ......... ........... ............. E24 2.00000 -1.15336 -0.66976 1.92583 -1.55142 -0.13648 1.70884 -1.83442 0.40691 1.36511 -1.98137 0.92013 0.92013 -1.98137 1.36511 0.40691 -1.83442 1.70884 -0.13648 -1.55142 1.92583 -0.66976 -1.15336 2.00000 -1.15336 -0.66976 1.92583 -1.55142 -0.13648 1.70884 -1.83442 0.40691 1.36511 -1.98137 0.92013 0.00000 ... ... ..... ....... ......... ........... ............. E25 2.00000 -1.29716 -0.31737 1.70884 -1.89928 0.75484 0.92013 -1.94840 1.60726 -0.13648 -1.43022 1.99171 -1.15336 -0.49562 1.79626 -1.83442 0.58329 1.07780 -1.98137 1.49237 0.04553 -1.55142 1.96692 -1.00000 -0.66976 1.86879 -1.75437 0.40691 1.22653 -1.99793 1.36511 0.22716 -1.65977 1.92583 -0.83835 0.00000 ... ... ..... ....... ......... ........... ............. E26 2.00000 -1.43022 0.04553 1.36511 -1.99793 1.49237 -0.13648 -1.29716 1.99171 -1.55142 0.22716 1.22653 -1.98137 1.60726 -0.31737 -1.15336 1.96692 -1.65977 0.40691 1.07780 -1.94840 1.70884 -0.49562 -1.00000 1.92583 -1.75437 0.58329 0.92013 -1.89928 1.79626 -0.66976 -0.83835 1.86879 -1.83442 0.75484 0.00000 ... ... ..... ....... ......... ........... ............. E27 2.00000 -1.55142 0.40691 0.92013 -1.83442 1.92583 -1.15336 -0.13648 1.36511 -1.98137 1.70884 -0.66976 -0.66976 1.70884 -1.98137 1.36511 -0.13648 -1.15336 1.92583 -1.83442 0.92013 0.40691 -1.55142 2.00000 -1.55142 0.40691 0.92013 -1.83442 1.92583 -1.15336 -0.13648 1.36511 -1.98137 1.70884 -0.66976 0.00000 ... ... ..... ....... ......... ........... ............. E28 2.00000 -1.65977 0.75484 0.40691 -1.43022 1.96692 -1.83442 1.07780 0.04553 -1.15336 1.86879 -1.94840 1.36511 -0.31737 -0.83835 1.70884 -1.99793 1.60726 -0.66976 -0.49562 1.49237 -1.98137 1.79626 -1.00000 -0.13648 1.22653 -1.89928 1.92583 -1.29716 0.22716 0.92013 -1.75437 1.99171 -1.55142 0.58329 0.00000 ... ... ..... ....... ......... ........... ............. E29 2.00000 -1.75437 1.07780 -0.13648 -0.83835 1.60726 -1.98137 1.86879 -1.29716 0.40691 0.58329 -1.43022 1.92583 -1.94840 1.49237 -0.66976 -0.31737 1.22653 -1.83442 1.99171 -1.65977 0.92013 0.04553 -1.00000 1.70884 -1.99793 1.79626 -1.15336 0.22716 0.75484 -1.55142 1.96692 -1.89928 1.36511 -0.49562 0.00000 ... ... ..... ....... ......... ........... ............. E30 2.00000 -1.83442 1.36511 -0.66976 -0.13648 0.92013 -1.55142 1.92583 -1.98137 1.70884 -1.15336 0.40691 0.40691 -1.15336 1.70884 -1.98137 1.92583 -1.55142 0.92013 -0.13648 -0.66976 1.36511 -1.83442 2.00000 -1.83442 1.36511 -0.66976 -0.13648 0.92013 -1.55142 1.92583 -1.98137 1.70884 -1.15336 0.40691 0.00000 ... ... ..... ....... ......... ........... ............. E31 2.00000 -1.89928 1.60726 -1.15336 0.58329 0.04553 -0.66976 1.22653 -1.65977 1.92583 -1.99793 1.86879 -1.55142 1.07780 -0.49562 -0.13648 0.75484 -1.29716 1.70884 -1.94840 1.99171 -1.83442 1.49237 -1.00000 0.40691 0.22716 -0.83835 1.36511 -1.75437 1.96692 -1.98137 1.79626 -1.43022 0.92013 -0.31737 0.00000 ... ... ..... ....... ......... ........... ............. E32 2.00000 -1.94840 1.79626 -1.55142 1.22653 -0.83835 0.40691 0.04553 -0.49562 0.92013 -1.29716 1.60726 -1.83442 1.96692 -1.99793 1.92583 -1.75437 1.49237 -1.15336 0.75484 -0.31737 -0.13648 0.58329 -1.00000 1.36511 -1.65977 1.86879 -1.98137 1.99171 -1.89928 1.70884 -1.43022 1.07780 -0.66976 0.22716 0.00000 ... ... ..... ....... ......... ........... ............. E33 2.00000 -1.98137 1.92583 -1.83442 1.70884 -1.55142 1.36511 -1.15336 0.92013 -0.66976 0.40691 -0.13648 -0.13648 0.40691 -0.66976 0.92013 -1.15336 1.36511 -1.55142 1.70884 -1.83442 1.92583 -1.98137 2.00000 -1.98137 1.92583 -1.83442 1.70884 -1.55142 1.36511 -1.15336 0.92013 -0.66976 0.40691 -0.13648 0.00000 ... ... ..... ....... ......... ........... ............. E34 2.00000 -1.99793 1.99171 -1.98137 1.96692 -1.94840 1.92583 -1.89928 1.86879 -1.83442 1.79626 -1.75437 1.70884 -1.65977 1.60726 -1.55142 1.49237 -1.43022 1.36511 -1.29716 1.22653 -1.15336 1.07780 -1.00000 0.92013 -0.83835 0.75484 -0.66976 0.58329 -0.49562 0.40691 -0.31737 0.22716 -0.13648 0.04553 0.00000 ... ... ..... ....... ......... ........... ............. Irrational character values: 1.991713685063 = 2*cos(2*π/69) 1.966923403267 = 2*cos(4*π/69) 1.925834574696 = 2*cos(6*π/69) = 2*cos(2*π/23) 1.868787674321 = 2*cos(8*π/69) 1.796255410727 = 2*cos(10*π/69) 1.708838809093 = 2*cos(12*π/69) = 2*cos(4*π/23) 1.607262230910 = 2*cos(14*π/69) 1.492367371695 = 2*cos(16*π/69) 1.365106286437 = 2*cos(18*π/69) = 2*cos(6*π/23) 1.226533500567 = 2*cos(20*π/69) 1.077797271831 = 2*cos(22*π/69) 0.920130075462 = 2*cos(24*π/69) = 2*cos(8*π/23) 0.754838391505 = 2*cos(26*π/69) 0.583291878910 = 2*cos(28*π/69) 0.406912026105 = 2*cos(30*π/69) = 2*cos(10*π/23) 0.227160372101 = 2*cos(32*π/69) 0.045526395712 = 2*cos(34*π/69) -0.136484826729 = 2*cos(36*π/69) = 2*cos(12*π/23) -0.317365092913 = 2*cos(38*π/69) -0.495615571986 = 2*cos(40*π/69) -0.669759224342 = 2*cos(42*π/69) = 2*cos(14*π/23) -0.838353040833 = 2*cos(44*π/69) -1.153360644230 = 2*cos(48*π/69) = 2*cos(16*π/23) -1.297164178926 = 2*cos(50*π/69) -1.430219002710 = 2*cos(52*π/69) -1.551422581409 = 2*cos(54*π/69) = 2*cos(18*π/23) -1.659770583998 = 2*cos(56*π/69) -1.754365204805 = 2*cos(58*π/69) -1.834422603011 = 2*cos(60*π/69) = 2*cos(20*π/23) -1.899279397801 = 2*cos(62*π/69) -1.948398165347 = 2*cos(64*π/69) -1.981371892073 = 2*cos(66*π/69) = 2*cos(22*π/23) -1.997927347293 = 2*cos(68*π/69) Symmetry of Rotations and Cartesian products A1 d+g+i+k+m z2, z4, z6 A2 R+p+f+h+j+l Rz, z, z3, z5 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 {x2−y2, xy}, {z(x2−y2), xyz}, {z2(x2−y2), xyz2}, {z3(x2−y2), xyz3}, {z4(x2−y2), xyz4} E3 f+g+h+i+j+k+l+m {x(x2−3y2), y(3x2−y2)}, {xz(x2−3y2), yz(3x2−y2)}, {xz2(x2−3y2), yz2(3x2−y2)}, {xz3(x2−3y2), yz3(3x2−y2)} E4 g+h+i+j+k+l+m {(x2−y2)2−4x2y2, xy(x2−y2)}, {z((x2−y2)2−4x2y2), xyz(x2−y2)}, {z2((x2−y2)2−4x2y2), xyz2(x2−y2)} E5 h+i+j+k+l+m {x(x2−(5+2√5)y2)(x2−(5−2√5)y2), y((5+2√5)x2−y2)((5−2√5)x2−y2)}, {xz(x2−(5+2√5)y2)(x2−(5−2√5)y2), yz((5+2√5)x2−y2)((5−2√5)x2−y2)} E6 i+j+k+l+m {x2(x2−3y2)2−y2(3x2−y2)2, xy(x2−3y2)(3x2−y2)} E7 j+k+l+m E8 k+l+m E9 l+m E10 m Notes: α The order of the D69 point group is 138, and the order of the principal axis (C69) is 69. The group has 36 irreducible representations. β The D69 point group is isomorphic to C69v. γ The D69 point group is generated by two symmetry elements, C69 and a perpendicular C2′. Also, the group may be generated from two C2′ axes (some pairs will yield smaller groups, though; choosing a minimum angle is safe). 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. ε The lowest nonvanishing multipole moment in D69 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 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 much less than half the order of the principal axis. For this group, however, none of the irrational characters can be expressed by a closed algebraic form using real numbers only.
| D67 | ||
| D68 | ||
| C69 C69v C69h | D69 | D69h D69d |
| D70 | ||
| D71 |
This Character Table for the D69 point group was created by Gernot Katzer.
For other groups and some explanations, see the Main Page.