Bound on the index of an abelian subgroup in discrete subgroup of the euclidean group? $\DeclareMathOperator{\isom}{Isom}$A discrete subgroup of the group of isometries in euclidean space is almost abelian.
By this I mean that for each $n$ there exists $m$ such that for any discrete subgroup $\Gamma$ of $\isom(\mathbb{R}^n)$ we can find an abelian subgroup $\Gamma' \leq \Gamma$ such that $$[\Gamma : \Gamma'] \leq m,$$ so the index of the abelian subgroup in $\Gamma$ is bounded by $m$.
What is the best known bound on $m$?

The index is the order of the crystallographic point group, in the cocompact case. 
It is then a finite subgroup of $GL(n,\mathbb{Z})$. There are several bounds known for the maximal order of such subgroups. Friedland proved in $1997$ that 
$$m\le 2^nn!
$$ 
for $n\ge n_0$ and gave conditions where equality is attained (The maximal orders of finite subgroups of $GL(n,\mathbb{Q})$.
Rockmore in $1995$ used a different description: For every $\epsilon>0$ there exists a constant $c(\epsilon)$ such that $m\le c(\epsilon)(n!)^{1+\epsilon}$.
