  We introduce the Hadamard full propelinear codes that factorize as direct
product of groups such that their associated group is $C_{2t}\times C_2$. We
study the rank, the dimension of the kernel, and the structure of these codes.
For several specific parameters we establish some links from circulant Hadamard
matrices and the nonexistence of the codes we study. We prove that the
dimension of the kernel of these codes is bounded by 3 if the code is
nonlinear. We also get an equivalence between circulant complex Hadamard matrix
and a type of Hadamard full propelinear code, and we find a new example of
circulant complex Hadamard matrix of order 16.
