Daisy chains with four generators

For many positive odd integers n, whether prime or not, the set U-n of units of Z(n) contains members t, u, v and w, say with respective orders tau, psi, omega and pi, such that we can write U-n as the direct product U-n - < t > x < u > x < v > x < w >. Each element of U-n can then be written in the form t(h)u(i)v(j)w(k) where 0 <= h < tau, 0 <= i < psi, 0 <= j < omega and 0 <= k < pi. We can then often use the structure of < t > x < u > x < v > x < w > to arrange the tau psi omega pi elements of U-n in a daisy chain, i.e. in a circular arrangement such that, as we proceed once round the chain in either direction, the set of differences between each member and the preceding one is itself the set U-n. We describe daisy chains based on such 4-factor decompositions as daisy chains with four generators. We study the existence of such arrangements, and we note their relationships with the previously studied daisy chains with three generators. The smallest prime values n for which daisy chains with four generators exist are 571 and 1051.