In this paper, we first show that the central height of the unit group of the integral group ring of a periodic group is at most $2$. We then give a complete characterization of the $n$-centre of that unit group. The $n$-centre of the unit group is either the centre or the second centre (for $n \geq 2$).

Iwahori-Hecke algebras for the infinite series of complex reflection groups $G(r,p,n)$ were constructed recently in the work of Ariki and Koike~\cite{AK}, Brou\'e and Malle \cite{BM}, and Ariki~\cite{Ari}. In this paper we give Murnaghan-Nakayama type formulas for computing the irreducible characters of these algebras. Our method is a generalization of that in our earlier paper ~\cite{HR} in which we derived Murnaghan-Nakayama rules for the characters of the Iwahori-Hecke algebras of the classical Weyl groups. In both papers we have been motivated by C. Greene~\cite{Gre}, who gave a new derivation of the Murnaghan-Nakayama formula for irreducible symmetric group characters by summing diagonal matrix entries in Young's seminormal representations. We use the analogous representations of the Iwahori-Hecke algebra of $G(r,p,n)$ given by Ariki and Koike~\cite{AK} and Ariki ~\cite{Ari}.