Represent the symmetry group of an equilateral triangle as a group of permutations of it...

Represent the symmetry group of an equilateral triangle as a group of permutations of its vertices (see Example 3).

Reference:

EXAMPLE 3 Symmetries of a Square As a third example, we associate each motion in D₄ with the permutation of the locations of each of the four corners of a square. For example, if we label the four corner positions as in the figure below and keep these labels fixed for reference, we may describe a 90° counterclockwise rotation by the permutation

whereas a reflection across a horizontal axis yields

These two elements generate the entire group (that is, every element is some combination of the ρ’s and ?’s).

When D₄ is represented in this way, we see that it is a subgroup of S₄.