Let $ G $ be a group of $ n \times n $ matrices. Suppose that some subset $ \{ g_j: 1 \leq j \leq n^2 \} $ of $ G $ is a basis for the space of all $ n \times n $ matrices. Furthermore suppose that the set $$ \{ \overline{g_i}: 1 \leq j \leq n^2 \} $$ is a group in $ \operatorname{PGL}_n $. What can we say about the size of $ G $? Must it be the case that $$ \lvert G\rvert\geq n^2 \prod p_i $$ where $ \prod p_i $ is the product of all distinct primes dividing $ n $?

When we take our field to be algebraically closed and characteristic 0 I think I have groups that saturate this bound for all $ n $. Is this bound even true? Can we do better? Mostly interested in doing everything over $ \mathbb{C} $ but could be cool to hear about other fields too.