  We consider the Wiener--Hopf factorization problem for a matrix function that
is completely defined by its first column: the succeeding columns are obtained
from the first one by means of a finite group of permutations. The symmetry of
this matrix function allows us to reduce the dimension of the problem. In
particular, we find some relations between its partial indices and can compute
some of the indices. In special cases we can explicitly obtain the Wiener--Hopf
factorization of the matrix function.
