In all these symmetries, alternate reflections can be removed producing the rotational tetrahedral [3,3] (), octahedral [3,4] (), and icosahedral [3,5] () groups of order 12, 24, and 60. The octahedral group also has a unique index 2 subgroup called the pyritohedral symmetry group, [3,4] ( or ), of order 12, with a mixture of rotational and reflectional symmetry. Pyritohedral symmetry is also an index 5 subgroup of icosahedral symmetry: --> , with virtual mirror 1 across 0, {010}, and 3-fold rotation {12}.