  The main objective of this paper is to compute $RO(G)$-graded cohomology of
$G$-orbits for the group $G=C_n$, where $n$ is a product of distinct primes. We
compute these groups for the constant Mackey functor $\underline{Z}$ and for
the Burnside ring Mackey functor $\underline{A}$. Among other things, we show
that the groups $\underline{H}^\alpha_G(S^0)$ are mostly determined by the
fixed point dimensions of the virtual representations $\alpha$, except in the
case of $\underline{A}$ coefficients when the fixed point dimensions of
$\alpha$ have many zeros. In the case of $\underline{Z}$ coefficients, the ring
structure on the cohomology is also described. The calculations are then used
to prove freeness results for certain $G$-complexes.
