In this question I ask for a generalization of [What is the maximal possible rank of a subgroup of a special linear group mod a prime?](https://mathoverflow.net/questions/219491/what-is-the-maximal-possible-rank-of-a-subgroup-of-a-special-linear-group-mod-a/219535#219535)


Let $p\_1, \dots, p\_r$ be $r$ distinct odd primes.


Set $$G = \prod\_{i=1}^r \mathrm{SL}\_3(\mathbb{F}\_{p\_i})$$ 


What is $\max \{d(H) : H \leq G\}$?


Here we denote by $d(K)$ the smallest cardinality of a generating set of a group $K$. 



The maximal possible rank of a subgroup of a product of special linear groups
I should answer the question! In my answer to the [earlier question](https://mathoverflow.net/questions/219491) I sketched a proof that the maximal rank of a subgroup of ${\rm SL}(3,p)$ with $p$ prime is $3$, except for when $p=2$, when it is $2$. So $3r$ is an upper bound on ranks of subgroups of the group $G$ in the question.


Now, when $p \equiv 1 \bmod 4$, ${\rm GL}(2,p)$ has an irreducible subgroup of rank $3$ equal to a central product of $Q\_8$ and a cyclic scalar subgroup of order $4$. Hence so does ${\rm SL}(3,p)$.


So, if we choose $p\_i \equiv 1 \bmod 4$ for $1 \le i \le r$, then the direct product of these $r$ rank $3$ subgroups is a $2$-subgroup of $G$ of rank $3r$.


Note that the largest proper subgroup of ${\rm SL}(3,p)$ is the parabolic subgroup $p^2:{\rm GL}(2,p)$, which has index $p^2+p+1$. Since this has a quotient of order $2$, the direct product of these subgroups is a subgroup of $G$ of rank (at least, and probably exactly) $r$ and index $\prod\_{i=1}^r (p\_i^2+p\_i+1)$.


I would guess that no subgroup of larger order has rank at least $r$, since any such subgroup would have to contain at least one of the quasisimple direct factors ${\rm SL}(3,p\_i)$


