Daisy chains with four generators

For many positive odd integers n, whether prime or not, the set Un of units of Zn contains members t, u, v and w, say with respective orders τ , ψ, ω and π, such that we can write Un as the direct product Un = 〈t〉 × 〈u〉 × 〈v〉 × 〈w〉. Each element of Un can then be written in the form tuvw where 0 ≤ h < τ , 0 ≤ i < ψ, 0 ≤ j < ω and 0 ≤ k < π. We can then often use the structure of 〈t〉 × 〈u〉 × 〈v〉 × 〈w〉 to arrange the τψωπ elements of Un 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 Un. 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.