One combinatorial construction in representation theory

Let K be an extension of the field Q_p of p-adic rationals, let O_K be its ring of integers, and let G be a finite group. According to the classical Jordan–Zassenhaus Theorem, if K∕Q_p is finite, every isomorphism class of KG-representation modules splits in a finite number of isomorphism classes of O_KG-representation modules. We consider a p-group G of a given nilpotency class k>1 and the extension K∕Q_p where K=Q_p(ζ_p^∞ ) obtained by adjoining all roots ζ_pⁱ ,i=1,2,3,. of unity, and we use an explicit combinatorial construction of a faithful absolutely irreducible KG-module M to show that the number of O_KG-isomorphism classes of O_KG-representation modules, which are KG-equivalent to M, is infinite, and there are extra congruence properties for these O_KG-modules.