conjugate p-elements of full support that generate the wreath product cp≀cp

For a symmetric group g:=symn>g:=symn and a conjugacy class x>x of involutions in g>g‎, ‎it is known that if the class of involutions does not have a unique fixed point‎, ‎then‎ - ‎with a few small exceptions‎ - ‎given two elements a,x∈x>a,x∈x‎, ‎either ⟨a,x⟩>⟨a,x⟩ is isomorphic to the dihedral group d8>d8‎, ‎or there is a further element y∈x>y∈x such that ⟨a,y⟩≅⟨x,y⟩≅d8>⟨a,y⟩≅⟨x,y⟩≅d8 (p‎. ‎rowley and d‎. ‎ward‎, ‎on π>π-product involution graphs in symmetric‎ ‎groups‎. ‎mims eprint‎, ‎2014)‎. ‎one natural generalisation of this to p>p-elements is to consider when two conjugate p>p-elements generate a wreath product of two cyclic groups of order p>p‎. ‎in this paper we give necessary and sufficient conditions for this in the case that our p>p-elements have full support‎. ‎these conditions relate to given matrices that are of circulant or permutation type‎, ‎and corresponding polynomials that represent these matrices‎. ‎we also consider the case that the elements do not have full support‎, ‎and see why generalising our results to such elements would not be a natural generalisation‎.