In the study of rotation symmetric Boolean functions (RSBFs), it is natural to consider the equivalence of Boolean vectors in F 2 n given by v∼ w if w is obtained from v by cyclic permutation (rotation). Several authors (Clark, Cusick, Hell, Maitra, Maximov, Stănică), in relation to RSBFs, considered the square matrix n A obtained as follows: let (G i) i= 1,…, g n be the equivalence classes of this relation∼ and Λ i be representatives; the entries of n A are (∑ x∈ G i (− 1) x⋅ Λ j) i, j. Some properties of this matrix were obtained for n odd in the literature. We obtain a few new formulas regarding the number of classes of various types, and investigate the matrix n A in general. One of our main results is that n A satisfies (n A) 2= 2 n Id, and it is conjugate to its transpose by a diagonal matrix. This is not an immediate consequence of the similar property of the related Hadamard type matrix (p v, w) v, w∈ F 2 n=((− 1) v⋅ w) v, w, but it is rather connected to character theory. We show that the entries of the matrix n A are essentially the character values of the irreducible representations of the semidirect (or wreath) product of F 2 n⋊ C n, where C n is the cyclic group with n elements, which yields further properties of this matrix. This connection suggests possible future investigations, and motivates the introduction of Boolean functions with various other types of symmetry.