  We define, for each subset $S$ of the set $\mathcal{P}$ of primes, an
$S_n$-module $Lie_n^S$ with interesting properties. $Lie_n^\emptyset$ is the
well-known representation $Lie_n$ of $S_n$ afforded by the free Lie algebra,
while $Lie_n^\mathcal{P}$ is the module $C\!onj_n$ of the conjugacy action of
$S_n$ on $n$-cycles. For arbitrary $S$ the module $Lie_n^{S}$ interpolates
between the representations $Lie_n$ and $C\!onj_n.$ We consider the symmetric
and exterior powers of $Lie_n^S.$ These are the analogues of the higher Lie
modules of Thrall. We show that the Frobenius characteristic of these higher
$Lie_n^S$ modules can be elegantly expressed as a multiplicity-free sum of
power sums. In particular this establishes the Schur positivity of new classes
of sums of power sums.
  More generally, for each nonempty subset $T$ of positive integers we define a
sequence of symmetric functions $f_n^T$ of homogeneous degree $n.$ We show that
the series $\sum_{\lambda, \lambda_i\in T} p_\lambda$ can be expressed as
symmetrised powers of the functions $f_n^T$, analogous to the higher Lie
modules first defined by Thrall. This in turn allows us to unify previous
results on the Schur positivity of multiplicity-free sums of power sums, as
well as investigate new ones. We also uncover some curious plethystic
relationships between $f_n^T$, the conjugacy action and the Lie representation.
