Let $G$ be a finite group, $n(G)$ the minimal number of generators and $m(G)$ the minimal number of irreducible complex representations generating (with $\otimes$ and $\oplus$) the left regular representation. 


*Notation*: the word "generating" does not mean "generating exactly", but as a direct factor.


*Question*: Is it true that $n(G) \ge m(G)$ ?


*Remark*: a group $G$ is linearly primitive iff $m(G) = 1$, so it's obviouly true in this case.



An inequality for the minimal number of generators of a finite group
Let $P$ be a $p$-group. Then the smallest number of irreducible characters of
$P$ whose kernels intersect trivially is at least $n(Z(P))$, and thus
$m(P) \ge n(Z(P))$. So to find an example where $n(P) < m(P)$, it suffices to find $P$ such that $n(P) < n(Z(P))$. An example in the Magma or GAP data
base of small groups is SmallGroup($3^6$,9). Here, $n(P) = 2$ but $n(Z(P)) = 3$.


