Most recent answer: 12/04/2015
I haven't ever studied the Weyl group so I ran this by my friend Mike Stone:
The generators are a pair of reflections in planes 120 dgrees apart. The product of these is a rotation through 2*120 =240 =-120 degrees. Its inverse is a rotation of +120. A conjugation gives another reflection in the third plane. Then there is the identity. I think that is 6 elements.
(published on 12/04/2015)