Cyclotomic Swan subgroups and primitive roots

Let Km=Q(ζm) where ζm is a primitive mth root of unity. Let p>2 be prime and let Cp denote the group of order p. The ring of algebraic integers of Km is Om=Z[ζm]. Let Λm,p denote the order Om[Cp] in the algebra Km[Cp]. Consider the kernel group D(Λm,p) and the Swan subgroup T(Λm,p). If (p,m)=1 these two subgroups of the class group coincide. Restricting to when there is a rational prime p that is prime in Om requires m=4 or qn where q>2 is prime. For each such m, 3⩽m⩽100, we give such a prime, and show that one may compute T(Λm,p) as a quotient of the group of units of a finite field. When hmp+=1 we give exact values for |T(Λm,p)|, and for other cases we provide an upper bound. We explore the Galois module theoretic implications of these results.