Conjugate $p$-elements of full support that generate the wreath product $C_{p}wr C_{p}$

For a symmetric group G:=symn">G:=symnG:=symn and a conjugacy class X">XX of involutions in G">GG‎, ‎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∈Xa,x∈X‎, ‎either ⟨a,x⟩">⟨a,x⟩⟨a,x⟩ is isomorphic to the dihedral group D8">D8D8‎, ‎or there is a further element y∈X">y∈Xy∈X such that ⟨a,y⟩≅⟨x,y⟩≅D8">⟨a,y⟩≅⟨x,y⟩≅D8⟨a,y⟩≅⟨x,y⟩≅D8 (P‎. ‎Rowley and D‎. ‎Ward‎, ‎On π">ππ-Product Involution Graphs in Symmetric‎ ‎Groups‎. ‎MIMS ePrint‎, ‎2014)‎.  ‎One natural generalisation of this to p">pp-elements is to consider when two conjugate p">pp-elements generate a wreath product of two cyclic groups of order p">pp‎. ‎In this paper we give necessary and sufficient conditions for this in the case that our p">pp-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.