Binary octahedral group
It follows that the binary octahedral group is a discrete subgroup of Spin(3) of order 48.
The binary octahedral group is most easily described concretely as a discrete subgroup of the unit quaternions, under the isomorphism
(For a description of this homomorphism see the article on quaternions and spatial rotations.)
Explicitly, the binary octahedral group is given as the union of the 24 Hurwitz units with all 24 quaternions obtained from by a permutation of coordinates and all possible sign combinations.
All 48 elements have absolute value 1 and therefore lie in the unit quaternion group Sp(1).
The binary tetrahedral group, 2T, consisting of the 24 Hurwitz units, forms a normal subgroup of index 2.
The quaternion group, Q8, consisting of the 8 Lipschitz units forms a normal subgroup of 2O of index 6.
All other subgroups are cyclic groups generated by the various elements (with orders 3, 4, 6, and 8).
• 1 order-1: 1
• 1 order-2: -1
• 6 order-4: ±i, ±j, ±k
• 12 order-8: (±1±i)/√2, (±1±j)/√2, (±1±k)/√2
• 12 order-4: (±i±j)/√2, (±i±k)/√2, (±j±k)/√2
• 8 order-6, (+1±i±j±k)/2
• 8 order-3, (-1±i±j±k)/2.
• 2 T =⟨2,3,3⟩, index 2,
• Q 16=⟨2,2,4⟩ index 3, and
• Q 12=⟨2,2,3⟩ index 4.
• ⟨ l , m , n ⟩= binary polyhedral group
• ⟨ p ⟩≃Z 2 p , ( p )≃Z p ( cyclic groups )