The dihedral group expresses the symmetries of a regular n-gons (n rotations
and n reflections so it is O(2n)).
D2= symmetries of a rectangle = {e,a,b,c} where a = reflection, b= reflection
through a different axis, and c=rotation by Pi.
a^2=e, b^2=e, c^2=e, ab=c=ba
All of the other relations between e,a,b,c should be derivable from this.
D3 is the simplest non-abelian group. It is the symmetries of a triangle.
D3= {e,c,c^2,b,bc,bc^2}
c= rotation by Pi/3 b= reflection through a fixed axis of reflection.
When applying transformations, apply the leftmost one first.
To see non-abelian consider:
(bc)^2=e=bcbc
=> b=cbc (b^2=e)
=> c^2b=bc (c^3=e)
=> non-abelian