The study of symmetry

I know that in mathematics, Group Theory is the formal study of symmetry. However, having done quite a lot of group theory I am left with this question:

How?

I understand that the dihedral groups, and subgroups thereof, represent symmetries wonderfully. But, for instance, what would be represented by the Symmetric groups? would it be some n-dimensional object (where the symmetric group is S_n)?

Originally posted by Swlabr
... would it be some n-dimensional object (where the symmetric group is S_n)?

You could be meaning: Sn is symmetric in the same way as a regular (n-1) simplex.

Originally posted by flexmore
You might have some grasp of group theory ... but your simple arithmetic has slipped just a little:

S3 is like a 2d triangle,
and
S4 is like a 3d cube.

Originally posted by flexmore
You might have some grasp of group theory ... but your simple arithmetic has slipped just a little:

S3 is like a 2d triangle,
and
S4 is like a 3d cube.