Voronoï ’s algorithm is a method for obtaining the complete list of perfect $n$-dimensional quadratic forms. Its generalization to $G$-forms has the advantage of running in a lower-dimensional space, and furnishes a finite, and complete, classification of $G$-perfect forms ($G$ is a finite subgroup of $GL(n,\mathbb{Z}))$. We study the standard, $\phi \left(m\right)$-dimensional irreducible representation of the cyclic group $C_m$ of order $m$, and give the, often new, densest $G$-forms. Perfect cyclotomic forms are completely classified for $\phi \left(m\right)\<16$ and for...