$\DeclareMathOperator\Rep{Rep}\DeclareMathOperator\rank{rank}$Let $G$ be a finite group, and $C=\Rep(G)$ be the monoidal category of complex finite-dimensional representations of $G$. As $C$ is finite and semisimple, one can get all representations from $\oplus$ and a finite set $I$ of irreducible representations. By classical character theory, there is a (noncanonical) bijection between $I$ and $\mathrm{Conj}(G)$. In this thread, I hope to understand a bijection, if any, between both sides with the consideration of $\otimes$.

To be more precise, let $V$ be an irreducible faithful representation of $G$. Then every representation occurs as a submodule of $V^{\otimes n}$ for some $n$ (cf this and this), and vice versa! We then say that $V$ itself generates $C$ under $\otimes$ and Cauchy completion. However, not every group has an irreducible faithful representation. In the same post, we can see that this largely deals with the "rank" of the socle of $G$.

To summarize, define the rank, $\rank(G)$, to be the minimal number of elements needed to generate $\mathrm{socle}(G)$ under conjugation. Define the rank, $\rank(C)$, to be the minimal number of irreducible elements needed to generate $C$ under $\otimes$ and Cauchy completion. Then

$$ \rank(G) = 1 \Leftrightarrow \rank(\Rep(G)) = 1 $$

### Question

Does this equivalence generalize to

$$ \rank(G) = n \Leftrightarrow \rank(\Rep(G)) = n, $$

for each natural number $n$?

(**EDIT** As Qiaochu pointed out in the comment, this is true for finite abelian groups by Pontrjagin duality.)