The ten plane crystallographic point symmetry groups

The general point groups described in Section 3.1 are not subject to any limitations. The point groups obtained by excluding all rotation operations incompatible with a lattice are called the crystallographic plane point groups. These are formed, therefore, by combining the rotation axes 1, 2, 3, 4, and 6, with mirror symmetry. When the 

The order of the symbols in the point group labels are allocated in the following way. The first (primary) position gives the rotation axis if present. The second (secondary) and third (tertiary) positions record whether a mirror element, m, is present. An m in the secondary position means that the mirror has a normal parallel to the [10] direction, in all lattices. If only one mirror is present it is always given with respect to this direction. An m in the secondary position has a normal parallel to [01] in a rectangular unit cell, and to [ll] in both a square and a hexagonal unit cell, (Table 3.2). 

For example, objects belonging to the point group 4mm, such as a square, have a tetrad axis, a mirror perpendicular to the [10] direction and an independent mirror perpendicular to the direction [11]. Objects belonging to the point group m, such as the letters A or C, have only a single mirror present. This would be perpendicular to, and thus define, the [10] direction in the object. 

To determine the crystallographic point group of a planar shape it is only necessary to write down a list of all of the symmetry elements present, order them following the rules set out above, and then compare them to the list of ten groups given. 

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