Let $\rho: G \to GL(V)$ be a finite (complex) group representation. What is the maximum dimension of $span\{\rho_gv | g \in G\}$ over all $v \in V$? This quantity is not necessarily the degree of $V$: for example, if $V$ is the direct sum of two isomorphic 1-dimensional subrepresentations, then the maximum dimension is 1.
I think this can be rephrased as: given a $\mathbb{C}G$-module, what is the largest (vector space) dimension over $\mathbb{C}$ of a cyclic submodule?
What I have so far: by the canonical decomposition, $V \cong W_1 \oplus W_2 \oplus ... \oplus W_r$ where each $W_i$ is isomorphic to $n_i$ copies of an irreducible submodule $U_i$ with dimension $d_i$. For $v \in V$ define $\Phi = \sum_{g \in G}{\rho_gvv^*\rho_g^*}$ (* denotes adjoint). Then the dimension quantity in question is equal to $rank(\Phi)$. $\Phi$ is a $\mathbb{C}G$-module homomorphism of $V$ into itself. Therefore $\Phi$ maps each $W_i$ to itself. Hence if $v = w_1 + w_2 + ... + w_r$ is the direct sum decomposition of $v$, then the dimension of $span\{\rho_g{v} | g \in G\} = \sum_{i = 1}^r{dim (span\{\rho_gw_i | g \in G\})}$, thus reducing the problem to the situation where the representation in question is isomorphic to copies of an irreducible module.
If this irreducible module has degree 1, then the maximum dimension is 1. But for degree > 1, I'm not sure. I haven't been formally taught representation theory, so I'd appreciate references as well as a clue to this problem.