Consider the following general problem. There is a finite group $G$ and $H_1,H_2 < G$. Suppose we know that $\langle H_1, H_2 \rangle = G$, i.e. $G$ is generated by $H_1$ and $H_2$. Denote by $n_0$ the minimal $n \in \mathbb{N}$ such that the following holds:
$$
G = \underbrace{H_1 H_2 H_1 H_2 \ldots H_1}_{\text{$2n-1$ subgroups}}
$$
**Problem:** *How to estimate $n_0$*?

This is, definitely, too general.

**Question:** *Do you know about any results of such kind regarding any specific group $G$ and corresponding specific $H_1$ and $H_2$? Maybe matrix groups?*