Euclidean group

These groups are among the oldest and most studied, at least in the cases of dimension 2 and 3 – implicitly, long before the concept of group was invented.

The Euclidean group is a subgroup of the group of affine transformations.

Countably infinite groups without arbitrarily small translations, rotations, or combinations
Countably infinite groups with arbitrarily small translations, rotations, or combinations
Non-countable groups, where there are points for which the set of images under the isometries is not closed
(e.g., in 2D all translations in one direction, and all translations by rational distances in another direction).
Non-countable groups, where for all points the set of images under the isometries is closed

E(1), E(2), and E(3) can be categorized as follows, with degrees of freedom:

The translations by a given distance in any direction form a conjugacy class; the translation group is the union of those for all distances.

In 2D, rotations by the same angle in either direction are in the same class. Glide reflections with translation by the same distance are in the same class.