The extended group [[4,3,4]] doubles the group order, adding with a 2-fold rotation matrix T, with a fixed axis through points (1,1/2,0) and (1/2,1/2,1/2). The generators are {R 0,R 1,T}. R 2 = T×R 1 ×T and R 3 = T×R 0 ×T.