on a die aaaa rotoreflection aaaa "grouped" group table

In the study of geometry, one is constantly confronted with groups of transformations on various "spaces." Many of these groups consist simply of the symmetries of those spaces with respect to suitably chosen properties. An obvious example is furnished by the symmetries of the cube. Geometrically speaking, these are the one-one transformations which preserve distances on the cube. They are known as "isometries," and are 48 in number.

[Birkhoff & MacLane, Survey of Modern Algebra (1941), p. 127]

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