Several chaotic properties of cyclic permutation maps are considered. Cyclic permutation maps refer to p-dimensional dynamical systems of the form φ (b 1 , b 2 , ⋯ , b p) = (u p (b p) , u 1 (b 1) , ⋯ , u p − 1 (b p − 1)) , where b j ∈ H j ( j ∈ { 1 , 2 , ⋯ , p } ), p ≥ 2 is an integer, and H j ( j ∈ { 1 , 2 , ⋯ , p } ) are compact subintervals of the real line R = (− ∞ , + ∞) . u j : H j → H j + 1 (j = 1 , 2 , ... , p − 1) and u p : H p → H 1 are continuous maps. Necessary and sufficient conditions for a class of cyclic permutation maps to have Li–Yorke chaos, distributional chaos in a sequence, distributional chaos, or Li–Yorke sensitivity are given. These results extend the existing ones. [ABSTRACT FROM AUTHOR]