# Diameter of finite rational matrix groups

Suppose $$G$$ is a finite subgroup of $$\mathrm{GL}(n,\mathbb{Q})$$. For a set $$\mathcal{M} \subseteq G$$ that generates $$G$$, define the $$\mathcal{M}$$-diameter $$\mathit{diam}(G, \mathcal{M})$$ of $$G$$ to be the smallest $$k \in \mathbb{N}$$ such that $$\bigcup_{i=0}^k \mathcal{M}^i = G$$, i.e., every matrix in $$G$$ can be written as a product of matrices from $$\mathcal{M}$$, with product length at most $$k$$. Further, define the diameter $$\mathit{diam}(G)$$ of $$G$$ as the maximum $$\mathcal{M}$$-diameter of $$G$$, over all sets $$\mathcal{M}$$ generating $$G$$. Define the function $$d : \mathbb{N} \to \mathbb{N}$$ such that $$d(n)$$ is the maximum $$\mathit{diam}(G)$$, over all finite subgroups $$G$$ of $$\mathrm{GL}(n,\mathbb{Q})$$.

Equivalently, $$d(n)$$ can be formally defined on one line by $$d(n) \ := \ \max \left\{\min\left\{k \in \mathbb{N} : \bigcup_{i=0}^k \mathcal{M}^i = \langle\mathcal{\mathcal{M}}\rangle\right\} : \langle\mathcal{\mathcal{M}}\rangle \le \mathrm{GL}(n,\mathbb{Q}) \text{ is finite}\right\}\,.$$

I am interested in lower and upper bounds on $$d(n)$$.

It is known from [S. Friedland. The maximal orders of finite subgroups in GL$$_n$$($$\mathbb{Q}$$). Proceedings of the American Mathematical Society, 125(12):3519-3526, 1997] that for large $$n$$ the maximal order of a finite subgroup of $$\mathrm{GL}(n,\mathbb{Q})$$ is $$2^n n!$$. Hence, $$d(n) \le 2^{C n \log n}$$ for some $$C>0$$.

On the other hand, denoting by $$p(n)$$ the maximal order of a permutation on $$n$$ elements, we have $$\lim_{n \to \infty} (\ln p(n))/{\sqrt{n \ln n}} = 1$$; see, e.g., [W. Miller. The maximum order of an element of a finite symmetric group. American Mathematical Monthly, 94(6):497-506, 1987]. Hence, $$d(n) \ge 2^{c \sqrt{n \log n}}$$ for some $$c>0$$.

Are there better bounds on $$d(n)$$?