Crystallographic group

A crystallographic group or space group $𝑆$ in dimension $𝑛$ is a discrete subgroup $𝑆$ of the Euclidean group $E (𝑛)$ containing a normal $ℝ 𝑛$ -lattice subgroup $𝐿 ⊴ 𝑆 \leq E (𝑛)$ such that quotient group $𝑆 / 𝐿$ , called the Point group, is a subgroup of the real orthogonal group $O (𝑛)$ . #m/def/group This gives the following pair of short exact sequences

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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