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..... ............. 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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 ... ... ..... ....... ......... ........... ............. 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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 {x2–y2, xy}, {z2(x2–y2), xyz2}, {z4(x2–y2), xyz4} E3 f+h {x(x2–3y2), y(3x2–y2)}, {xz2(x2–3y2), yz2(3x2–y2)} E4 g+i {(x2–y2)2–2x2y2, xy(x2–y2)}, {z2((x2–y2)2–4x2y2), xyz2(x2–y2)} 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)2–y2(3x2–y2)2, xy(x2–3y2)(3x2–y2)} 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((x2–y2)2–4x2y2), xyz(x2–y2)} E53 g+i {xz(x2–3y2), yz(3x2–y2)}, {xz3(x2–3y2), yz3(3x2–y2)} E54 f+h {z(x2–y2), xyz}, {z3(x2–y2), 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) = (3√28+i*84*√3 + 3√28-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.
S110 | ||
C112 C112v C112h D112 D112h D112d | S112 | |
S114 |
This Character Table for the S112 point group was created by Gernot Katzer.
For other groups and some explanations, see the Main Page.