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In mathematics, the binary tetrahedral group, denoted 2T or ⟨2,3,3⟩, is a certain nonabelian group of order 24. It is an extension of the tetrahedral group T or of order 12 by a cyclic group of order 2, and is the preimage of the tetrahedral group under the 2:1 covering homomorphism Spin → SO of the special orthogonal group by the spin group. It follows that the binary tetrahedral group is a discrete subgroup of Spin of order 24. The complex reflection group named 33 by G.C. Shephard or 33 and by Coxeter, is isomorphic to the binary tetrahedral group.
The binary tetrahedral group is most easily described concretely as a discrete subgroup of the unit quaternions, under the isomorphism Spin ≅ Sp, where Sp is the multiplicative group of unit quaternions.