[Formerly: Publ. I.R.M.A. Strasbourg, 1986, 316/S-13, p. 135-144.]

where *P* represents the permutation (1 2 ... *n*),
(*n,g*)=1, and the exponents are mod *n*.

We prove that if (*k,n*)=*h*, a matrix *Q* is
*k*-quasi-*g*-circulant if and only if it is
*h*-quasi-*g*-circulant; then *Q* is a block
*g*-circulant matrix of type (*q,h*), and we give a
characterization for these matrices. Moreover, we define a perfect
*k*-quasi-*g*-circulant permutation, and we prove that the
set of these permutations is an imprimitive group of order
*phi*(*k*)*kq ^{k}*, where

The main part of the results of this paper have appeared in the article "On the solutions of a matrix equation,"