  In this paper an approach to generate multi-dimensionally consistent
$N$-component systems is proposed. The approach starts from scalar
multi-dimensionally consistent quadrilateral systems and makes use of the
cyclic group. The obtained $N$-component systems inherit integrable features
such as B\"acklund transformations and Lax pairs, and exhibit interesting
aspects, such as nonlocal reductions. Higher order single component lattice
equations (on larger stencils) and multi-component discrete Painlev\'e
equations can also be derived in the context, and the approach extends to
$N$-component generalizations of higher dimensional lattice equations.
