the combination, where 2, 3, 6, , and axes are present, axes , , and 6 all generate 6-point patterns, as we can see on the figure in the right, but the latter should be used because it is a rotation axis — the symbol will be . Finally, the Hermann–Mauguin symbol depends on the type of the group. Groups without higher-order axes (axes of order three or more) These groups may contain only two-fold axes, mirror planes, and/or an inversion center. These are the crystallographic point groups 1 and (triclinic crystal system), 2, m, and (monoclinic), and 222, , and mm2 (orthorhombic). (The short form of is mmm.) If the symbol contains three positions, then they denote symmetry elements in the x, y, z direction, respectively. Groups with one higher-order axis First position — primary direction — z direction, assigned to the higher-order axis. Second position — symmetrically equivalent secondary directions, which are perpendicular to the z-axis. These can be 2, m, or . Third position — symmetrically equivalent tertiary directions, passing between secondary directions. These can be 2, m, or . These are the crystallographic groups 3, 32, 3m, , and (trigonal crystal system), 4, 422, 4mm, , 2m, , and (tetragonal), and 6, 622, 6mm, , m2, , and (hexagonal). Analogously, symbols of non-crystallographic groups (with axes of order 5, 7, 8, 9 ...) can be constructed. These groups can be arranged in the following table It can be noticed that in groups with odd-order axes n and the third position in symbol is always absent, because all n directions, perpendicular to higher-order axis, are symmetrically equivalent. For example, in the picture of a triangle all three mirror planes (S0, S1, S2) are equivalent — all of them pass through one vertex and the center of the opposite side. For even-order axes n and there are secondary directions and tertiary directions. For example, in the picture of a regular hexagon one can distinguish two sets of mirror planes — three planes go through two opposite vertexes, and three other planes go through the centers of opposite sides. In this case any of two sets can be chosen as secondary directions, the rest set will be tertiary directions. Hence groups 2m, 2m, 2m, ... can be written as m2, m2, m2, ... . For symbols of point groups this order usually doesn't matter; however, it will be important