A brief summary of plane groups

The symbols for plane groups, the Hermann-Mauguin symbol, have been the standard in crystallography. The first place indicates the type of lattice, p indicates primitive, and c indicates centered. The second place indicates the axial symmetry, which has only 5 possible vales, 1-, 2-, 3-, 4-, and 6-fold. For the rest, the letter m indicates a symmetry under a mirror reflection, and the letter g indicates a symmetry with respect to a glide line, that is, one-half of the unit vector translation followed by a mirror reflection. For example, the plane group pAmm means that the surface has fourfold symmetry as well as mirror reflection symmetries through both x and y axes. 

The 17 plane groups are not mutually unrelated. Some of them are subgroups of other plane groups, as shown in Fig. E.l. The order of the factor group, that is, the number of different symmetry operations other than translational symmetry, is also shown for each plane group. 

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