On the radical idealizer chain of symmetric orders

If Λ is an indecomposable, non maximal, symmetric order, then the idealizer of the radical Γ := Id(J(Λ)) = J(Λ)# is the dual of the radical. If Γ is hereditary then Λ has a Brauer tree (under modest additional assumptions). Otherwise ∆ := Id(J(Γ)) = (J(Γ)2)#. If Λ = ZpG for a p-group G 6= 1, then Γ is hereditary iff G ∼= Cp and otherwise [∆ : Λ] = p2|G/(G′Gp)|. For Abelian groups G, the length of the radical idealizer chain of ZpG is (n−a)(pa−pa−1)+pa−1, where p is the order and p the exponent of the Sylow p-subgroup of G.