We introduce the CPF-group, the group whose elements consist of flavor and CP symmetries. We elucidate that the peculiarities of these symmetries are due to the fact that transformation matrices acting on fields, do not form a matrix representation of the CPF-group. To construct such matrices, we use the standard method that considers how the automorphism group of the flavor group is acting on the corresponding transformation matrices. We also introduce a new method to obtain transformation matrices that only uses real representations of the CPF-group.