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Every discrete group of Euclidean isometries acts properly discontinuously and cocompactly on some subspace of $\mathbb{R}^n$ (a result of Bieberbach). If we are in the crystallographic case, then the index is bounded by the maximal order of a finite subgroup of $GL(n,\mathbb{Z})$. Here we have several results in the literature, e.g., the result of Friedland in $1997$, that $$m\le 2^nn! $$ for $n\ge n_0$. There are exceptions for certain $n$, see the comment of Geoff Robinson. There are conditions given in Friedland's paper "The maximal orders of finite subgroups of $GL(n,\mathbb{Q})$", when equality is attained. Rockmore in $1995$ proved the following: for every $\epsilon>0$ there exists a constant $c(\epsilon)$ such that $m\le c(\epsilon)(n!)^{1+\epsilon}$.

Every discrete group of Euclidean isometries acts properly discontinuously and cocompactly on some subspace of $\mathbb{R}^n$ (a result of Bieberbach). If we are in the crystallographic case, then the index is bounded by the maximal order of a finite subgroup of $GL(n,\mathbb{Z})$. Here we have several results in the literature, e.g., the result of Friedland in $1997$, that $$m\le 2^nn! $$ for $n\ge n_0$. There are exceptions for small $n$, see the comment of Geoff Robinson. There are conditions given in Friedland's paper "The maximal orders of finite subgroups of $GL(n,\mathbb{Q})$", when equality is attained. Rockmore in $1995$ proved the following: for every $\epsilon>0$ there exists a constant $c(\epsilon)$ such that $m\le c(\epsilon)(n!)^{1+\epsilon}$.

Every discrete group of Euclidean isometries acts cocompactly on some subspace of $\mathbb{R}^n$ (a result of Bieberbach). If we are in the crystallographic case, then the index is bounded by the maximal order of a finite subgroup of $GL(n,\mathbb{Z})$. Here we have several results in the literature, e.g., the result of Friedland in $1997$, that $$m\le 2^nn! $$ for $n\ge n_0$. There are exceptions for certain $n$, see the comment of Geoff Robinson. There are conditions given in Friedland's paper "The maximal orders of finite subgroups of $GL(n,\mathbb{Q})$", when equality is attained. Rockmore in $1995$ proved the following: for every $\epsilon>0$ there exists a constant $c(\epsilon)$ such that $m\le c(\epsilon)(n!)^{1+\epsilon}$.

Every discrete group of Euclidean isometries acts properly discontinuously and cocompactly on some subspace of $\mathbb{R}^n$ (a result of Bieberbach). If we are in the crystallographic case, then the index is bounded by the maximal order of a finite subgroup of $GL(n,\mathbb{Z})$. Here we have several results in the literature, e.g., the result of Friedland in $1997$, that $$m\le 2^nn! $$ for $n\ge n_0$. There are exceptions for certain $n$, see the comment of Geoff Robinson. There are conditions given in Friedland's paper "The maximal orders of finite subgroups of $GL(n,\mathbb{Q})$", when equality is attained. Rockmore in $1995$ proved the following: for every $\epsilon>0$ there exists a constant $c(\epsilon)$ such that $m\le c(\epsilon)(n!)^{1+\epsilon}$.

Since every discrete group of Euclidean isometries acts cocompactly on some subspace of $\mathbb{R}^n$ (a result of Bieberbach), we may assume that we are in the crystallographic case, i.e., the index is bounded by the maximal order of a finite subgroup of $GL(n,\mathbb{Z})$. Here we have several results in the literature, e.g., the result of Friedland in $1997$, that $$m\le 2^nn! $$ for $n\ge n_0$. There are conditions given in Friedland's paper "The maximal orders of finite subgroups of $GL(n,\mathbb{Q})$", when equality is attained. Rockmore in $1995$ proved the following: for every $\epsilon>0$ there exists a constant $c(\epsilon)$ such that $m\le c(\epsilon)(n!)^{1+\epsilon}$.

Every discrete group of Euclidean isometries acts cocompactly on some subspace of $\mathbb{R}^n$ (a result of Bieberbach). If we are in the crystallographic case, then the index is bounded by the maximal order of a finite subgroup of $GL(n,\mathbb{Z})$. Here we have several results in the literature, e.g., the result of Friedland in $1997$, that $$m\le 2^nn! $$ for $n\ge n_0$. There are exceptions for certain $n$, see the comment of Geoff Robinson. There are conditions given in Friedland's paper "The maximal orders of finite subgroups of $GL(n,\mathbb{Q})$", when equality is attained. Rockmore in $1995$ proved the following: for every $\epsilon>0$ there exists a constant $c(\epsilon)$ such that $m\le c(\epsilon)(n!)^{1+\epsilon}$.