 Finite-dimensional representation obstruction from elementary abelian subgroups

Commuting complex involutions are simultaneously diagonalizable, so $(C_2{)}^n$ can act faithfully in dimension $d$ only if $n\le d$. A group containing $(C_2{)}^n$ for arbitrarily large $n$, such as the full permutation group of $\mathbb{N}$, has no faithful finite-dimensional complex representation.