We explore the *CPT* group in a manner that is not typically found in quantum mechanics textbooks, revealing a larger structure than is commonly known. After a brief review of the restricted Poincaré group, we derive the irreducible representations of single-particle quantum mechanical states of the discrete elements of the extended Poincaré group adjoined by charge conjugation. We show that there are 16 different irreducible representations of the discrete operations of charge conjugation $C,$ space inversion $P,$ and time reversal $T.$ The familiar conventional representations are identified and interpretations of the unconventional irreducible representations that lead to a doubling of the space of states under time reversal are discussed.

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