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