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:<math>\chi^J (2\pi+\alpha) = -\chi^J (\alpha)</math>
The change of sign cannot be true for an identity operation in any point group. Therefore, a double group, in which rotation by <math> 2\pi</math>, is classified as being distinct from the identity operation, is used.
A character table for the double group ''D'''<sub>4</sub> is as follows.
The new The character table for the point group ''D''<sub>4</sub> is shown for comparison.
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In
Character tables for the double groups T', O', T<sub>d</sub>', D<sub>3h</sub>', C<sub>6v</sub>', D<sub>6</sub>', D<sub>2d</sub>', C<sub>4v</sub>', D<sub>4</sub>', C<sub>3v</sub>', D<sub>3</sub>', C<sub>2v</sub>', D<sub>2</sub>' and R(3)' are given in Salthouse and Ware.<ref name=sw>{{cite book |last1=Salthouse |first1=J.A. |last2=Ware |first2=M.J. |title=Point group character tables and related data |date=1972 |publisher=[[Cambridge University Press]] |location=Cambridge |isbn=0 521 081394 |pages=55-57}}</ref>
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