Properties
The binary octahedral group, denoted by 2O, fits into the short exact sequence
This sequence does not split, meaning that 2O is not a semidirect product of {±1} by O. In fact, there is no subgroup of 2O isomorphic to O.
The center of 2O is the subgroup {±1}, so that the inner automorphism group is isomorphic to O. The full automorphism group is isomorphic to O × Z2.
Read more about this topic: Binary Octahedral Group
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