Icosahedral Groups
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The icosahedral groups may be derived by trial and error as in the Using SYMGRP section, but of course it is easier to use the known relations between the symmetry operators (as supplied by Franco Zefiro):

Smallest angle between 5-fold and 3-fold axes = 37.37736814°

Smallest angle between 5-fold and 2-fold axes = 31.71747440°

Smallest angle between 3-fold and 2-fold axes = 20.90515744°

SYMGRP is programmed to generate a 5-fold axis automatically using the first angle - see Symmetry Operator Dialog.

Here is one way to make icosahedral groups.

1) Insert a 3-fold axis parallel to x.

2) Insert a 5-fold axis using the special icosahedral option in the Symmetry Operator Dialog. The first axis, to which the 3-fold axis is parallel, is x and the second axis is y.

3) (Optional) To get 2-fold axes parallel to x, y and z and a 3-fold in the [111] position, as in cubic crystals, in the Operator List Dialog rotate by 20.90515744 degrees on z. Add the center of inversion if desired. Now calculate and save the .sym file.