The Symmetric group of the set ${1,2...,n}$ is $S_n$. It is the set of permuations of the set ${1,2...,n}$.
But what is the symmetry group of a polygon? or a $3D$ shape?
For example I saw:
"A cuboid has distinct dimensions-its symmetry group is $C_2$x$C_2$x$C_2$".
What is meant by symmetry group in this context and how is it "$C_2$x$C_2$x$C_2$". It then says:
"When two (but not three) of the dimensions are the same then the symmetry group is now $D_8$"...?
On
The symmetry group (and not symmetric group, which involves permutations) of a polygon is the group of (geometric) actions (or transformations) which leave the polygon the same (invariant).
For example, the symmetry group for a square (dihedral group), includes rotations of $90^o$, reflections (or flips) along vertical/horizontal/diagonal axes and so on..
I think you are confusing symmetric group and group of symmetry. The latter are more commonly called dihedral groups hence the notation $D_8$.
Dihedral groups consist of rotations and flips of the polygon.