Note on E representations in
C30 character table



28 irreducible representations of point group C30 have complex values. 14 two-dimensional real-valued representations E can be constructed as direct sum of the 14 pairs complex plus conjugate complex irreducible representation.

E1 = E1a ⊕ E1b
E2 = E2a ⊕ E2b
E3 = E3a ⊕ E3b
E4 = E4a ⊕ E4b
E5 = E5a ⊕ E5b
E6 = E6a ⊕ E6b
E7 = E7a ⊕ E7b
E8 = E8a ⊕ E8b
E9 = E9a ⊕ E9b
E10 = E10a ⊕ E10b
E11 = E11a ⊕ E11b
E12 = E12a ⊕ E12b
E13 = E13a ⊕ E13b
E14 = E14a ⊕ E14b


ε=exp(2πi/30)
C30 E C30 C15 C10 (C15)2 C6 C5 (C30)7 (C15)4 (C10)3 C3 (C30)11 (C5)2 (C30)13 (C15)7 C2 (C15)8 (C30)17 (C5)3 (C30)19 (C3)2 (C10)7 (C15)11 (C30)23 (C5)4 (C6)5 (C15)13 (C10)9 (C15)14 (C30)29
A 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
B 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 E1a
E1b
1
1
ε*
ε*
ε2*
ε2*
ε3*
ε3*
ε4*
ε4*
ε5*
ε5*
ε6*
ε6*
ε7*
ε7*
7*
7*
6*
6*
5*
5*
4*
4*
3*
3*
2*
2*
*
*
-1
-1
*
*
2*
2*
3*
3*
4*
4*
5*
5*
6*
6*
7*
7*
ε7*
ε7*
ε6*
ε6*
ε5*
ε5*
ε4*
ε4*
ε3*
ε3*
ε2*
ε2*
ε*
ε*
E2 E2a
E2b
1
1
ε2*
ε2*
ε4*
ε4*
ε6*
ε6*
7*
7*
5*
5*
3*
3*
*
*
*
*
3*
3*
5*
5*
7*
7*
ε6*
ε6*
ε4*
ε4*
ε2*
ε2*
1
1
ε2*
ε2*
ε4*
ε4*
ε6*
ε6*
7*
7*
5*
5*
3*
3*
*
*
*
*
3*
3*
5*
5*
7*
7*
ε6*
ε6*
ε4*
ε4*
ε2*
ε2*
E3 E3a
E3b
1
1
ε3*
ε3*
ε6*
ε6*
6*
6*
3*
3*
-1
-1
3*
3*
6*
6*
ε6*
ε6*
ε3*
ε3*
1
1
ε3*
ε3*
ε6*
ε6*
6*
6*
3*
3*
-1
-1
3*
3*
6*
6*
ε6*
ε6*
ε3*
ε3*
1
1
ε3*
ε3*
ε6*
ε6*
6*
6*
3*
3*
-1
-1
3*
3*
6*
6*
ε6*
ε6*
ε3*
ε3*
E4 E4a
E4b
1
1
ε4*
ε4*
7*
7*
3*
3*
*
*
5*
5*
ε6*
ε6*
ε2*
ε2*
ε2*
ε2*
ε6*
ε6*
5*
5*
*
*
3*
3*
7*
7*
ε4*
ε4*
1
1
ε4*
ε4*
7*
7*
3*
3*
*
*
5*
5*
ε6*
ε6*
ε2*
ε2*
ε2*
ε2*
ε6*
ε6*
5*
5*
*
*
3*
3*
7*
7*
ε4*
ε4*
E5 E5a
E5b
1
1
ε5*
ε5*
5*
5*
-1
-1
5*
5*
ε5*
ε5*
1
1
ε5*
ε5*
5*
5*
-1
-1
5*
5*
ε5*
ε5*
1
1
ε5*
ε5*
5*
5*
-1
-1
5*
5*
ε5*
ε5*
1
1
ε5*
ε5*
5*
5*
-1
-1
5*
5*
ε5*
ε5*
1
1
ε5*
ε5*
5*
5*
-1
-1
5*
5*
ε5*
ε5*
E6 E6a
E6b
1
1
ε6*
ε6*
3*
3*
3*
3*
ε6*
ε6*
1
1
ε6*
ε6*
3*
3*
3*
3*
ε6*
ε6*
1
1
ε6*
ε6*
3*
3*
3*
3*
ε6*
ε6*
1
1
ε6*
ε6*
3*
3*
3*
3*
ε6*
ε6*
1
1
ε6*
ε6*
3*
3*
3*
3*
ε6*
ε6*
1
1
ε6*
ε6*
3*
3*
3*
3*
ε6*
ε6*
E7 E7a
E7b
1
1
ε7*
ε7*
*
*
6*
6*
ε2*
ε2*
ε5*
ε5*
3*
3*
4*
4*
ε4*
ε4*
ε3*
ε3*
5*
5*
2*
2*
ε6*
ε6*
ε*
ε*
7*
7*
-1
-1
7*
7*
ε*
ε*
ε6*
ε6*
2*
2*
5*
5*
ε3*
ε3*
ε4*
ε4*
4*
4*
3*
3*
ε5*
ε5*
ε2*
ε2*
6*
6*
*
*
ε7*
ε7*
E8 E8a
E8b
1
1
7*
7*
*
*
ε6*
ε6*
ε2*
ε2*
5*
5*
3*
3*
ε4*
ε4*
ε4*
ε4*
3*
3*
5*
5*
ε2*
ε2*
ε6*
ε6*
*
*
7*
7*
1
1
7*
7*
*
*
ε6*
ε6*
ε2*
ε2*
5*
5*
3*
3*
ε4*
ε4*
ε4*
ε4*
3*
3*
5*
5*
ε2*
ε2*
ε6*
ε6*
*
*
7*
7*
E9 E9a
E9b
1
1
6*
6*
3*
3*
ε3*
ε3*
ε6*
ε6*
-1
-1
ε6*
ε6*
ε3*
ε3*
3*
3*
6*
6*
1
1
6*
6*
3*
3*
ε3*
ε3*
ε6*
ε6*
-1
-1
ε6*
ε6*
ε3*
ε3*
3*
3*
6*
6*
1
1
6*
6*
3*
3*
ε3*
ε3*
ε6*
ε6*
-1
-1
ε6*
ε6*
ε3*
ε3*
3*
3*
6*
6*
E10 E10a
E10b
1
1
5*
5*
5*
5*
1
1
5*
5*
5*
5*
1
1
5*
5*
5*
5*
1
1
5*
5*
5*
5*
1
1
5*
5*
5*
5*
1
1
5*
5*
5*
5*
1
1
5*
5*
5*
5*
1
1
5*
5*
5*
5*
1
1
5*
5*
5*
5*
1
1
5*
5*
5*
5*
E11 E11a
E11b
1
1
4*
4*
7*
7*
ε3*
ε3*
*
*
ε5*
ε5*
ε6*
ε6*
2*
2*
ε2*
ε2*
6*
6*
5*
5*
ε*
ε*
3*
3*
ε7*
ε7*
ε4*
ε4*
-1
-1
ε4*
ε4*
ε7*
ε7*
3*
3*
ε*
ε*
5*
5*
6*
6*
ε2*
ε2*
2*
2*
ε6*
ε6*
ε5*
ε5*
*
*
ε3*
ε3*
7*
7*
4*
4*
E12 E12a
E12b
1
1
3*
3*
ε6*
ε6*
ε6*
ε6*
3*
3*
1
1
3*
3*
ε6*
ε6*
ε6*
ε6*
3*
3*
1
1
3*
3*
ε6*
ε6*
ε6*
ε6*
3*
3*
1
1
3*
3*
ε6*
ε6*
ε6*
ε6*
3*
3*
1
1
3*
3*
ε6*
ε6*
ε6*
ε6*
3*
3*
1
1
3*
3*
ε6*
ε6*
ε6*
ε6*
3*
3*
E13 E13a
E13b
1
1
2*
2*
ε4*
ε4*
6*
6*
7*
7*
ε5*
ε5*
3*
3*
ε*
ε*
*
*
ε3*
ε3*
5*
5*
ε7*
ε7*
ε6*
ε6*
4*
4*
ε2*
ε2*
-1
-1
ε2*
ε2*
4*
4*
ε6*
ε6*
ε7*
ε7*
5*
5*
ε3*
ε3*
*
*
ε*
ε*
3*
3*
ε5*
ε5*
7*
7*
6*
6*
ε4*
ε4*
2*
2*
E14 E14a
E14b
1
1
*
*
ε2*
ε2*
3*
3*
ε4*
ε4*
5*
5*
ε6*
ε6*
7*
7*
7*
7*
ε6*
ε6*
5*
5*
ε4*
ε4*
3*
3*
ε2*
ε2*
*
*
1
1
*
*
ε2*
ε2*
3*
3*
ε4*
ε4*
5*
5*
ε6*
ε6*
7*
7*
7*
7*
ε6*
ε6*
5*
5*
ε4*
ε4*
3*
3*
ε2*
ε2*
*
*


Obviously the E representations are reducible. Nevertheless the E representations are treated often as irreducible representations and are called real-valued or pseudo irreducible representations. One should keep in mind that general statements for character tables fail for real-valued representations. For example:



Last update November, 13th 2023 by A. Gelessus, Impressum, Datenschutzerklärung/DataPrivacyStatement