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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. Wecompute these groups for the constant Mackey functor $\underline{Z}$ and forthe Burnside ring Mackey functor $\underline{A}$. Among other things, we showthat the groups $\underline{H}^\alpha_G(S^0)$ are mostly determined by thefixed point dimensions of the virtual representations $\alpha$, except in thecase of $\underline{A}$ coefficients when the fixed point dimensions of$\alpha$ have many zeros. In the case of $\underline{Z}$ coefficients, the ringstructure on the cohomology is also described. The calculations are then usedto prove freeness results for certain $G$-complexes.

Equivariant cohomology for cyclic groups of square-free order