This paper concerns the class of
finite groups whose complex irreducible character degrees can be linearly ordered by
divisibility. It is known that such a group has a Sylow tower. By analyzing the
structure of a group in the class whose order is divisible by just two primes, we are
able to obtain information on the Sylow subgroups of any group in the class. We
classify the two-prime groups in the class Except for certain exceptional pairs of
primes, one of which is always 2. Such a group G of order paqb either has a
normal abelian Sylow q-subgroup or H = G∕Op(G) has a non-abelian Sylow
q-subgroup Q and each p-element of H induces a fixedpoint-free automorphism of
Q′.