There don't seem to be any simple ways of expressing SU(3) values and higher, just as for SO(5) and higher.
I'll now take on the finite subgroups of O(n) and SO(n) for small n. For n = 1, rotations and reflections of a line (1-space), it is very easy.
For n = 2 (2-space or plane), the symmetry groups for a p-sided regular polygon are C(p) (cyclic) for pure rotations and D(p) ("dihedral") for rotations and reflections. I have some demos of these groups in these pages:
- C(p) contains all rotation angles 2pi*{0, 1/p, 2/p, ..., (p-1)/p}
- D(p) contains C(p) and reflections with reflection lines separated by angle pi/p.
As an example, let's consider a square. Its symmetry group is D(4), with rotation angles 0d, 90d, 180d, and 270d, and reflection lines parallel to pairs of sides and along the two diagonals.
Turn it into a rectangle. Its symmetry goes down to D(2), with rotation angles 0d and 180d, and reflection lines parallel to pairs of sides.
Then a parallelogram. Its symmetry goes down to C(2) with rotation angles 0d and 180d.
Alternately, then a trapezoid with the inclined sides inclined by the same amount. Its symmetry goes down to D(1) with rotation angle 0d and a reflection line between the two tilted sides.
No symmetry is C(1), rotation angle 0d -- the identity group.