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Ok, I have notice that for several finite groups the following situation occurs... I will use the non-abelian group of order 27 to illustrate the point I'm making:

The group has 11 charachers, 9 of which are linear.

The group has derived subgroup G' (= Z(G) the centre of the group...irrelevant!) has 3 elements, G/G' is isomorphic to C_3 x C_3

If the non linear characters are called Chi_10 and Chi_11, why are they equal to zero on G/G'?

Another example of where this occurs would be A_4, which has 3 linear characters, and one non-llinear, and the derived subgroup= V_4 so Chi_4, the non-linear character = 0 on the conjugacy classes (123) and (132)

Thank You very much!