Point Group D28d



D28d E 2S56 2C28 2(S56)3 2C14 2(S56)5 2(C28)3 2S8 2C7 2(S56)9 2(C28)5 2(S56)11 2(C14)3 2(S56)13 2C4 2(S56)15 2(C7)2 2(S56)17 2(C28)9 2(S56)19 2(C14)5 2(S8)3 2(C28)11 2(S56)23 2(C7)3 2(S56)25 2(C28)13 2(S56)27 C2 28C'2 28σd
A1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
A2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 -1 -1
B1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 1 -1
B2 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1
E1 2 1.9874 1.9499 1.8878 1.8019 1.6934 1.5637 1.4142 1.2470 1.0641 0.8678 0.6606 0.4450 0.2239 0 -0.2239 -0.4450 -0.6606 -0.8678 -1.0641 -1.2470 -1.4142 -1.5637 -1.6934 -1.8019 -1.8878 -1.9499 -1.9874 -2 0 0
E2 2 1.9499 1.8019 1.5637 1.2470 0.8678 0.4450 0 -0.4450 -0.8678 -1.2470 -1.5637 -1.8019 -1.9499 -2 -1.9499 -1.8019 -1.5637 -1.2470 -0.8678 -0.4450 0 0.4450 0.8678 1.2470 1.5637 1.8019 1.9499 2 0 0
E3 2 1.8878 1.5637 1.0641 0.4450 -0.2239 -0.8678 -1.4142 -1.8019 -1.9874 -1.9499 -1.6934 -1.2470 -0.6606 0 0.6606 1.2470 1.6934 1.9499 1.9874 1.8019 1.4142 0.8678 0.2239 -0.4450 -1.0641 -1.5637 -1.8878 -2 0 0
E4 2 1.8019 1.2470 0.4450 -0.4450 -1.2470 -1.8019 -2 -1.8019 -1.2470 -0.4450 0.4450 1.2470 1.8019 2 1.8019 1.2470 0.4450 -0.4450 -1.2470 -1.8019 -2 -1.8019 -1.2470 -0.4450 0.4450 1.2470 1.8019 2 0 0
E5 2 1.6934 0.8678 -0.2239 -1.2470 -1.8878 -1.9499 -1.4142 -0.4450 0.6606 1.5637 1.9874 1.8019 1.0641 0 -1.0641 -1.8019 -1.9874 -1.5637 -0.6606 0.4450 1.4142 1.9499 1.8878 1.2470 0.2239 -0.8678 -1.6934 -2 0 0
E6 2 1.5637 0.4450 -0.8678 -1.8019 -1.9499 -1.2470 0 1.2470 1.9499 1.8019 0.8678 -0.4450 -1.5637 -2 -1.5637 -0.4450 0.8678 1.8019 1.9499 1.2470 0 -1.2470 -1.9499 -1.8019 -0.8678 0.4450 1.5637 2 0 0
E7 2 1.4142 0 -1.4142 -2 -1.4142 0 1.4142 2 1.4142 0 -1.4142 -2 -1.4142 0 1.4142 2 1.4142 0 -1.4142 -2 -1.4142 0 1.4142 2 1.4142 0 -1.4142 -2 0 0
E8 2 1.2470 -0.4450 -1.8019 -1.8019 -0.4450 1.2470 2 1.2470 -0.4450 -1.8019 -1.8019 -0.4450 1.2470 2 1.2470 -0.4450 -1.8019 -1.8019 -0.4450 1.2470 2 1.2470 -0.4450 -1.8019 -1.8019 -0.4450 1.2470 2 0 0
E9 2 1.0641 -0.8678 -1.9874 -1.2470 0.6606 1.9499 1.4142 -0.4450 -1.8878 -1.5637 0.2239 1.8019 1.6934 0 -1.6934 -1.8019 -0.2239 1.5637 1.8878 0.4450 -1.4142 -1.9499 -0.6606 1.2470 1.9874 0.8678 -1.0641 -2 0 0
E10 2 0.8678 -1.2470 -1.9499 -0.4450 1.5637 1.8019 0 -1.8019 -1.5637 0.4450 1.9499 1.2470 -0.8678 -2 -0.8678 1.2470 1.9499 0.4450 -1.5637 -1.8019 0 1.8019 1.5637 -0.4450 -1.9499 -1.2470 0.8678 2 0 0
E11 2 0.6606 -1.5637 -1.6934 0.4450 1.9874 0.8678 -1.4142 -1.8019 0.2239 1.9499 1.0641 -1.2470 -1.8878 0 1.8878 1.2470 -1.0641 -1.9499 -0.2239 1.8019 1.4142 -0.8678 -1.9874 -0.4450 1.6934 1.5637 -0.6606 -2 0 0
E12 2 0.4450 -1.8019 -1.2470 1.2470 1.8019 -0.4450 -2 -0.4450 1.8019 1.2470 -1.2470 -1.8019 0.4450 2 0.4450 -1.8019 -1.2470 1.2470 1.8019 -0.4450 -2 -0.4450 1.8019 1.2470 -1.2470 -1.8019 0.4450 2 0 0
E13 2 0.2239 -1.9499 -0.6606 1.8019 1.0641 -1.5637 -1.4142 1.2470 1.6934 -0.8678 -1.8878 0.4450 1.9874 0 -1.9874 -0.4450 1.8878 0.8678 -1.6934 -1.2470 1.4142 1.5637 -1.0641 -1.8019 0.6606 1.9499 -0.2239 -2 0 0
E14 2 0 -2 0 2 0 -2 0 2 0 -2 0 2 0 -2 0 2 0 -2 0 2 0 -2 0 2 0 -2 0 2 0 0
E15 2 -0.2239 -1.9499 0.6606 1.8019 -1.0641 -1.5637 1.4142 1.2470 -1.6934 -0.8678 1.8878 0.4450 -1.9874 0 1.9874 -0.4450 -1.8878 0.8678 1.6934 -1.2470 -1.4142 1.5637 1.0641 -1.8019 -0.6606 1.9499 0.2239 -2 0 0
E16 2 -0.4450 -1.8019 1.2470 1.2470 -1.8019 -0.4450 2 -0.4450 -1.8019 1.2470 1.2470 -1.8019 -0.4450 2 -0.4450 -1.8019 1.2470 1.2470 -1.8019 -0.4450 2 -0.4450 -1.8019 1.2470 1.2470 -1.8019 -0.4450 2 0 0
E17 2 -0.6606 -1.5637 1.6934 0.4450 -1.9874 0.8678 1.4142 -1.8019 -0.2239 1.9499 -1.0641 -1.2470 1.8878 0 -1.8878 1.2470 1.0641 -1.9499 0.2239 1.8019 -1.4142 -0.8678 1.9874 -0.4450 -1.6934 1.5637 0.6606 -2 0 0
E18 2 -0.8678 -1.2470 1.9499 -0.4450 -1.5637 1.8019 0 -1.8019 1.5637 0.4450 -1.9499 1.2470 0.8678 -2 0.8678 1.2470 -1.9499 0.4450 1.5637 -1.8019 0 1.8019 -1.5637 -0.4450 1.9499 -1.2470 -0.8678 2 0 0
E19 2 -1.0641 -0.8678 1.9874 -1.2470 -0.6606 1.9499 -1.4142 -0.4450 1.8878 -1.5637 -0.2239 1.8019 -1.6934 0 1.6934 -1.8019 0.2239 1.5637 -1.8878 0.4450 1.4142 -1.9499 0.6606 1.2470 -1.9874 0.8678 1.0641 -2 0 0
E20 2 -1.2470 -0.4450 1.8019 -1.8019 0.4450 1.2470 -2 1.2470 0.4450 -1.8019 1.8019 -0.4450 -1.2470 2 -1.2470 -0.4450 1.8019 -1.8019 0.4450 1.2470 -2 1.2470 0.4450 -1.8019 1.8019 -0.4450 -1.2470 2 0 0
E21 2 -1.4142 0 1.4142 -2 1.4142 0 -1.4142 2 -1.4142 0 1.4142 -2 1.4142 0 -1.4142 2 -1.4142 0 1.4142 -2 1.4142 0 -1.4142 2 -1.4142 0 1.4142 -2 0 0
E22 2 -1.5637 0.4450 0.8678 -1.8019 1.9499 -1.2470 0 1.2470 -1.9499 1.8019 -0.8678 -0.4450 1.5637 -2 1.5637 -0.4450 -0.8678 1.8019 -1.9499 1.2470 0 -1.2470 1.9499 -1.8019 0.8678 0.4450 -1.5637 2 0 0
E23 2 -1.6934 0.8678 0.2239 -1.2470 1.8878 -1.9499 1.4142 -0.4450 -0.6606 1.5637 -1.9874 1.8019 -1.0641 0 1.0641 -1.8019 1.9874 -1.5637 0.6606 0.4450 -1.4142 1.9499 -1.8878 1.2470 -0.2239 -0.8678 1.6934 -2 0 0
E24 2 -1.8019 1.2470 -0.4450 -0.4450 1.2470 -1.8019 2 -1.8019 1.2470 -0.4450 -0.4450 1.2470 -1.8019 2 -1.8019 1.2470 -0.4450 -0.4450 1.2470 -1.8019 2 -1.8019 1.2470 -0.4450 -0.4450 1.2470 -1.8019 2 0 0
E25 2 -1.8878 1.5637 -1.0641 0.4450 0.2239 -0.8678 1.4142 -1.8019 1.9874 -1.9499 1.6934 -1.2470 0.6606 0 -0.6606 1.2470 -1.6934 1.9499 -1.9874 1.8019 -1.4142 0.8678 -0.2239 -0.4450 1.0641 -1.5637 1.8878 -2 0 0
E26 2 -1.9499 1.8019 -1.5637 1.2470 -0.8678 0.4450 0 -0.4450 0.8678 -1.2470 1.5637 -1.8019 1.9499 -2 1.9499 -1.8019 1.5637 -1.2470 0.8678 -0.4450 0 0.4450 -0.8678 1.2470 -1.5637 1.8019 -1.9499 2 0 0
E27 2 -1.9874 1.9499 -1.8878 1.8019 -1.6934 1.5637 -1.4142 1.2470 -1.0641 0.8678 -0.6606 0.4450 -0.2239 0 0.2239 -0.4450 0.6606 -0.8678 1.0641 -1.2470 1.4142 -1.5637 1.6934 -1.8019 1.8878 -1.9499 1.9874 -2 0 0
View A
View B
✅


Additional information

Number of symmetry elements h = 112
Number of classes, irreps n = 31
Abelian group no
Optical Isomerism (Chirality) no
Polar no
Parity no


Reduce representation to irreducible representations


E 2S56 2C28 2(S56)3 2C14 2(S56)5 2(C28)3 2S8 2C7 2(S56)9 2(C28)5 2(S56)11 2(C14)3 2(S56)13 2C4 2(S56)15 2(C7)2 2(S56)17 2(C28)9 2(S56)19 2(C14)5 2(S8)3 2(C28)11 2(S56)23 2(C7)3 2(S56)25 2(C28)13 2(S56)27 C2 28C'2 28σd



Genrate representation from irreducible representations


A1 A2 B1 B2 E1 E2 E3 E4 E5 E6 E7 E8 E9 E10 E11 E12 E13 E14 E15 E16 E17 E18 E19 E20 E21 E22 E23 E24 E25 E26 E27



Direct products of irreducible representations




Error: Reduction failed

Reduction to integer numbers for irreducible representations failed.
Error Code: 1

Binary products
⊙ A1 A2 B1 B2 E1 E2 E3 E4 E5 E6 E7 E8 E9 E10 E11 E12 E13 E14 E15 E16 E17 E18 E19 E20 E21 E22 E23 E24 E25 E26 E27
A1
Characters of reducible representation
E 2S56 2C28 2(S56)3 2C14 2(S56)5 2(C28)3 2S8 2C7 2(S56)9 2(C28)5 2(S56)11 2(C14)3 2(S56)13 2C4 2(S56)15 2(C7)2 2(S56)17 2(C28)9 2(S56)19 2(C14)5 2(S8)3 2(C28)11 2(S56)23 2(C7)3 2(S56)25 2(C28)13 2(S56)27 C2 28C'2 28σd
5 21.746 32.387 4.806 5.453 2.795 2.571 27.485 19.411 48.473 23.518 36.971 7.513 8.890 1 0.964 22.477 16.649 46.917 24.624 41.034 10.515 12.984 0.108 0.112 17.574 13.623 44.400 25 1 1