Point Groups

In geometry, a point group is a mathematical group of symmetry operations (isometries in a Euclidean space) that have a fixed point in common. The coordinate origin of the Euclidean space is conventionally taken to be a fixed point, and every point group in dimension $d$ is then a subgroup of the orthogonal group $O(d)$.

Each point group can be represented as sets of orthogonal matrices $M$ that transform a point $x$ into point $y$ by $y=Mx$. Each element is either a rotation ($det(M)=1$) or a reflection / improper rotation ($det(M)=-1$) .

Chiral and achiral point groups, reflection groups

Point groups can be classified into chiral (purely rotational) groups and achiral groups. Chiral groups are subgroups of the special orthogonal group $SO(d)$, only orthogonal transformations of determinant $+1$. Achiral groups allow for determinant $-1$.

Crystallographic Point Groups

A crystallographic point group is a $d=3$ point group whose symmetry operations are compatible with a three-dimensional lattice. According to the crystallographic restriction it may only contain one-, two-, three-, four-, and sixfold rotations or rotoinversions (rotation + reflection). This reduces the number of crystallographic point groups from an $\infty$ of general point groups to 32.

To each space group is associated a crystallographic point group by forgetting the translational components of the symmetry operations.

Notation

Schoenflies notation

In this notation, point groups are denoted by a letter symbol with a subscript

• $C_n$ (for cyclic)
  • group that has an n-fold rotation axis
• $C_{nh}$
  • $C_n$ + reflection plane perpendicular to axis of rotation
• $C_{nv}$
  • $C_n$ + mirror planes parallel to axis of rotation
• $S_{2n}$ (for Spiegel, German for mirror)
  • group with only a 2n-fold rotation-reflection axis
• $D_n$ (for dihedral, or two-sided)
  • group that has an n-fold rotation axis + n twofold axes perpendicular to that axis
• $D_{nh}$
  • $D_n$ + mirror plane perpendicular to $n$-fold axis
• $D_{nd}$
  • $D_n$ + mirror planes parallel to $n$-fold axis

References

• Point group wikipedia
• Crystallographic point groups wikipedia