Note on E representations in
C17h character table



32 irreducible representations of point group C17h have complex values. 16 two-dimensional real-valued representations E can be constructed as direct sum of the 16 pairs complex plus conjugate complex irreducible representation.

E'1 = E'1a ⊕ E'1b
E'2 = E'2a ⊕ E'2b
E'3 = E'3a ⊕ E'3b
E'4 = E'4a ⊕ E'4b
E'5 = E'5a ⊕ E'5b
E'6 = E'6a ⊕ E'6b
E'7 = E'7a ⊕ E'7b
E'8 = E'8a ⊕ E'8b
E''1 = E''1a ⊕ E''1b
E''2 = E''2a ⊕ E''2b
E''3 = E''3a ⊕ E''3b
E''4 = E''4a ⊕ E''4b
E''5 = E''5a ⊕ E''5b
E''6 = E''6a ⊕ E''6b
E''7 = E''7a ⊕ E''7b
E''8 = E''8a ⊕ E''8b


ε=exp(2πi/17)
C17h E C17 (C17)2 (C17)3 (C17)4 (C17)5 (C17)6 (C17)7 (C17)8 (C17)9 (C17)10 (C17)11 (C17)12 (C17)13 (C17)14 (C17)15 (C17)16 σh S17 (S17)19 (S17)3 (S17)21 (S17)5 (S17)23 (S17)7 (S17)25 (S17)9 (S17)27 (S17)11 (S17)29 (S17)13 (S17)31 (S17)15 (S17)33
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 1 1 1 1
E'1 E'1a
E'1b
1
1
ε*
ε*
ε2*
ε2*
ε3*
ε3*
ε4*
ε4*
ε5*
ε5*
ε6*
ε6*
ε7*
ε7*
ε8*
ε8*
ε8*
ε8*
ε7*
ε7*
ε6*
ε6*
ε5*
ε5*
ε4*
ε4*
ε3*
ε3*
ε2*
ε2*
ε*
ε*
1
1
ε*
ε*
ε2*
ε2*
ε3*
ε3*
ε4*
ε4*
ε5*
ε5*
ε6*
ε6*
ε7*
ε7*
ε8*
ε8*
ε8*
ε8*
ε7*
ε7*
ε6*
ε6*
ε5*
ε5*
ε4*
ε4*
ε3*
ε3*
ε2*
ε2*
ε*
ε*
E'2 E'2a
E'2b
1
1
ε2*
ε2*
ε4*
ε4*
ε6*
ε6*
ε8*
ε8*
ε7*
ε7*
ε5*
ε5*
ε3*
ε3*
ε*
ε*
ε*
ε*
ε3*
ε3*
ε5*
ε5*
ε7*
ε7*
ε8*
ε8*
ε6*
ε6*
ε4*
ε4*
ε2*
ε2*
1
1
ε2*
ε2*
ε4*
ε4*
ε6*
ε6*
ε8*
ε8*
ε7*
ε7*
ε5*
ε5*
ε3*
ε3*
ε*
ε*
ε*
ε*
ε3*
ε3*
ε5*
ε5*
ε7*
ε7*
ε8*
ε8*
ε6*
ε6*
ε4*
ε4*
ε2*
ε2*
E'3 E'3a
E'3b
1
1
ε3*
ε3*
ε6*
ε6*
ε8*
ε8*
ε5*
ε5*
ε2*
ε2*
ε*
ε*
ε4*
ε4*
ε7*
ε7*
ε7*
ε7*
ε4*
ε4*
ε*
ε*
ε2*
ε2*
ε5*
ε5*
ε8*
ε8*
ε6*
ε6*
ε3*
ε3*
1
1
ε3*
ε3*
ε6*
ε6*
ε8*
ε8*
ε5*
ε5*
ε2*
ε2*
ε*
ε*
ε4*
ε4*
ε7*
ε7*
ε7*
ε7*
ε4*
ε4*
ε*
ε*
ε2*
ε2*
ε5*
ε5*
ε8*
ε8*
ε6*
ε6*
ε3*
ε3*
E'4 E'4a
E'4b
1
1
ε4*
ε4*
ε8*
ε8*
ε5*
ε5*
ε*
ε*
ε3*
ε3*
ε7*
ε7*
ε6*
ε6*
ε2*
ε2*
ε2*
ε2*
ε6*
ε6*
ε7*
ε7*
ε3*
ε3*
ε*
ε*
ε5*
ε5*
ε8*
ε8*
ε4*
ε4*
1
1
ε4*
ε4*
ε8*
ε8*
ε5*
ε5*
ε*
ε*
ε3*
ε3*
ε7*
ε7*
ε6*
ε6*
ε2*
ε2*
ε2*
ε2*
ε6*
ε6*
ε7*
ε7*
ε3*
ε3*
ε*
ε*
ε5*
ε5*
ε8*
ε8*
ε4*
ε4*
E'5 E'5a
E'5b
1
1
ε5*
ε5*
ε7*
ε7*
ε2*
ε2*
ε3*
ε3*
ε8*
ε8*
ε4*
ε4*
ε*
ε*
ε6*
ε6*
ε6*
ε6*
ε*
ε*
ε4*
ε4*
ε8*
ε8*
ε3*
ε3*
ε2*
ε2*
ε7*
ε7*
ε5*
ε5*
1
1
ε5*
ε5*
ε7*
ε7*
ε2*
ε2*
ε3*
ε3*
ε8*
ε8*
ε4*
ε4*
ε*
ε*
ε6*
ε6*
ε6*
ε6*
ε*
ε*
ε4*
ε4*
ε8*
ε8*
ε3*
ε3*
ε2*
ε2*
ε7*
ε7*
ε5*
ε5*
E'6 E'6a
E'6b
1
1
ε6*
ε6*
ε5*
ε5*
ε*
ε*
ε7*
ε7*
ε4*
ε4*
ε2*
ε2*
ε8*
ε8*
ε3*
ε3*
ε3*
ε3*
ε8*
ε8*
ε2*
ε2*
ε4*
ε4*
ε7*
ε7*
ε*
ε*
ε5*
ε5*
ε6*
ε6*
1
1
ε6*
ε6*
ε5*
ε5*
ε*
ε*
ε7*
ε7*
ε4*
ε4*
ε2*
ε2*
ε8*
ε8*
ε3*
ε3*
ε3*
ε3*
ε8*
ε8*
ε2*
ε2*
ε4*
ε4*
ε7*
ε7*
ε*
ε*
ε5*
ε5*
ε6*
ε6*
E'7 E'7a
E'7b
1
1
ε7*
ε7*
ε3*
ε3*
ε4*
ε4*
ε6*
ε6*
ε*
ε*
ε8*
ε8*
ε2*
ε2*
ε5*
ε5*
ε5*
ε5*
ε2*
ε2*
ε8*
ε8*
ε*
ε*
ε6*
ε6*
ε4*
ε4*
ε3*
ε3*
ε7*
ε7*
1
1
ε7*
ε7*
ε3*
ε3*
ε4*
ε4*
ε6*
ε6*
ε*
ε*
ε8*
ε8*
ε2*
ε2*
ε5*
ε5*
ε5*
ε5*
ε2*
ε2*
ε8*
ε8*
ε*
ε*
ε6*
ε6*
ε4*
ε4*
ε3*
ε3*
ε7*
ε7*
E'8 E'8a
E'8b
1
1
ε8*
ε8*
ε*
ε*
ε7*
ε7*
ε2*
ε2*
ε6*
ε6*
ε3*
ε3*
ε5*
ε5*
ε4*
ε4*
ε4*
ε4*
ε5*
ε5*
ε3*
ε3*
ε6*
ε6*
ε2*
ε2*
ε7*
ε7*
ε*
ε*
ε8*
ε8*
1
1
ε8*
ε8*
ε*
ε*
ε7*
ε7*
ε2*
ε2*
ε6*
ε6*
ε3*
ε3*
ε5*
ε5*
ε4*
ε4*
ε4*
ε4*
ε5*
ε5*
ε3*
ε3*
ε6*
ε6*
ε2*
ε2*
ε7*
ε7*
ε*
ε*
ε8*
ε8*
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 -1 -1 -1 -1
E''1 E''1a
E''1b
1
1
ε*
ε*
ε2*
ε2*
ε3*
ε3*
ε4*
ε4*
ε5*
ε5*
ε6*
ε6*
ε7*
ε7*
ε8*
ε8*
ε8*
ε8*
ε7*
ε7*
ε6*
ε6*
ε5*
ε5*
ε4*
ε4*
ε3*
ε3*
ε2*
ε2*
ε*
ε*
-1
-1
*
*
2*
2*
3*
3*
4*
4*
5*
5*
6*
6*
7*
7*
8*
8*
8*
8*
7*
7*
6*
6*
5*
5*
4*
4*
3*
3*
2*
2*
*
*
E''2 E''2a
E''2b
1
1
ε2*
ε2*
ε4*
ε4*
ε6*
ε6*
ε8*
ε8*
ε7*
ε7*
ε5*
ε5*
ε3*
ε3*
ε*
ε*
ε*
ε*
ε3*
ε3*
ε5*
ε5*
ε7*
ε7*
ε8*
ε8*
ε6*
ε6*
ε4*
ε4*
ε2*
ε2*
-1
-1
2*
2*
4*
4*
6*
6*
8*
8*
7*
7*
5*
5*
3*
3*
*
*
*
*
3*
3*
5*
5*
7*
7*
8*
8*
6*
6*
4*
4*
2*
2*
E''3 E''3a
E''3b
1
1
ε3*
ε3*
ε6*
ε6*
ε8*
ε8*
ε5*
ε5*
ε2*
ε2*
ε*
ε*
ε4*
ε4*
ε7*
ε7*
ε7*
ε7*
ε4*
ε4*
ε*
ε*
ε2*
ε2*
ε5*
ε5*
ε8*
ε8*
ε6*
ε6*
ε3*
ε3*
-1
-1
3*
3*
6*
6*
8*
8*
5*
5*
2*
2*
*
*
4*
4*
7*
7*
7*
7*
4*
4*
*
*
2*
2*
5*
5*
8*
8*
6*
6*
3*
3*
E''4 E''4a
E''4b
1
1
ε4*
ε4*
ε8*
ε8*
ε5*
ε5*
ε*
ε*
ε3*
ε3*
ε7*
ε7*
ε6*
ε6*
ε2*
ε2*
ε2*
ε2*
ε6*
ε6*
ε7*
ε7*
ε3*
ε3*
ε*
ε*
ε5*
ε5*
ε8*
ε8*
ε4*
ε4*
-1
-1
4*
4*
8*
8*
5*
5*
*
*
3*
3*
7*
7*
6*
6*
2*
2*
2*
2*
6*
6*
7*
7*
3*
3*
*
*
5*
5*
8*
8*
4*
4*
E''5 E''5a
E''5b
1
1
ε5*
ε5*
ε7*
ε7*
ε2*
ε2*
ε3*
ε3*
ε8*
ε8*
ε4*
ε4*
ε*
ε*
ε6*
ε6*
ε6*
ε6*
ε*
ε*
ε4*
ε4*
ε8*
ε8*
ε3*
ε3*
ε2*
ε2*
ε7*
ε7*
ε5*
ε5*
-1
-1
5*
5*
7*
7*
2*
2*
3*
3*
8*
8*
4*
4*
*
*
6*
6*
6*
6*
*
*
4*
4*
8*
8*
3*
3*
2*
2*
7*
7*
5*
5*
E''6 E''6a
E''6b
1
1
ε6*
ε6*
ε5*
ε5*
ε*
ε*
ε7*
ε7*
ε4*
ε4*
ε2*
ε2*
ε8*
ε8*
ε3*
ε3*
ε3*
ε3*
ε8*
ε8*
ε2*
ε2*
ε4*
ε4*
ε7*
ε7*
ε*
ε*
ε5*
ε5*
ε6*
ε6*
-1
-1
6*
6*
5*
5*
*
*
7*
7*
4*
4*
2*
2*
8*
8*
3*
3*
3*
3*
8*
8*
2*
2*
4*
4*
7*
7*
*
*
5*
5*
6*
6*
E''7 E''7a
E''7b
1
1
ε7*
ε7*
ε3*
ε3*
ε4*
ε4*
ε6*
ε6*
ε*
ε*
ε8*
ε8*
ε2*
ε2*
ε5*
ε5*
ε5*
ε5*
ε2*
ε2*
ε8*
ε8*
ε*
ε*
ε6*
ε6*
ε4*
ε4*
ε3*
ε3*
ε7*
ε7*
-1
-1
7*
7*
3*
3*
4*
4*
6*
6*
*
*
8*
8*
2*
2*
5*
5*
5*
5*
2*
2*
8*
8*
*
*
6*
6*
4*
4*
3*
3*
7*
7*
E''8 E''8a
E''8b
1
1
ε8*
ε8*
ε*
ε*
ε7*
ε7*
ε2*
ε2*
ε6*
ε6*
ε3*
ε3*
ε5*
ε5*
ε4*
ε4*
ε4*
ε4*
ε5*
ε5*
ε3*
ε3*
ε6*
ε6*
ε2*
ε2*
ε7*
ε7*
ε*
ε*
ε8*
ε8*
-1
-1
8*
8*
*
*
7*
7*
2*
2*
6*
6*
3*
3*
5*
5*
4*
4*
4*
4*
5*
5*
3*
3*
6*
6*
2*
2*
7*
7*
*
*
8*
8*


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 August, 12th 2020 by A. Gelessus, Impressum, Datenschutzerklärung/DataPrivacyStatement