[[Condensed matter physics MOC]]
# Crystallographic group

A **crystallographic group** or **space group** $S$ in dimension $n$ is a [[discrete subgroup]] $S$ of the [[Euclidean group]] $\mathrm{E}(n)$ containing a [[Normal subgroup|normal]] $\mathbb{R}^n$-[[lattice subgroup]] $L \trianglelefteq S \leq \mathrm{E}(n)$
such that [[quotient group]] $S / L$, called the [[Point group]], is a subgroup of the [[real orthogonal group]] $\mathrm{O}(n)$. #m/def/group 
This gives the following pair of [[Short exact sequence|short exact sequences]]

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Such a group is the symmetry group of a [[Crystal]].
According to the [[Crystallographic restriction theorem]], point groups must have rotational symmetries that are 1-,2-,3-,4-, or 6-fold.

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