  Using the standard representation of the Dirac equation we show that, up to
signs, there exist only TWO SETS of consistent solutions for the matrices of
charge conjugation (C), parity (P), and time reversal (T). In both cases,
P^2=-1, and then two succesive applications of the parity transformation to
spin 1/2 fields NECESSARILY amounts to a 2\pi rotation. Each of these sets
generates a non abelian group of sixteen elements, G_1 and G_2, which are non
isomorphic subgroups of the Dirac algebra, which, being a Clifford algebra,
gives a geometric nature to the generators, in particular to C. It turns out
that G_1 is isomorphic to D_8 x Z_2, where D_8 is the dihedral group of eight
elements (the symmetries of the square) and Z_2 is isomorphic to S^0 (the
0-sphere); while G_2 is isomorphic to a certain semidirect product of D_8 and
Z_2. Instead, the corresponding quantum operators for C, P, and T generate a
UNIQUE group G, which we call THE CPT GROUP OF THE DIRAC FIELD, and which is
isomorphic to Q x Z_2, where Q is the quaternion group. This group, however, is
compatible only with the second of the above two matrix solutions, namely with
G_2, which is then called THE MATRIX CPT GROUP. It is interesting to remark
that G_1, G_2, and G are the only non abelian groups of sixteen elements with
three generators. Finally, the matrix groups are also given in the Weyl and
Majorana representations, suitable for taking the massless limit and for
describing self-conjugate fields.
