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

Hadamard full propelinear codes with associated group $C_{2t}\times C_2$; rank and kernel