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Concept
Binary tetrahedral group
Summary
In mathematics, the binary tetrahedral group, denoted 2T or , is a certain nonabelian group of order 24. It is an extension of the tetrahedral group T or (2,3,3) of order 12 by a cyclic group of order 2, and is the of the tetrahedral group under the 2:1 covering homomorphism Spin(3) → SO(3) of the special orthogonal group by the spin group. It follows that the binary tetrahedral group is a discrete subgroup of Spin(3) of order 24. The complex reflection group named 3(24)3 by G.C. Shephard or 3[3]3 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(3) ≅ Sp(1), where Sp(1) is the multiplicative group of unit quaternions. (For a description of this homomorphism see the article on quaternions and spatial rotations.)
Explicitly, the binary tetrahedral group is given as the group of units in the ring of Hurwitz integers. There are 24 such units given by
with all possible sign combinations.
All 24 units have absolute value 1 and therefore lie in the unit quaternion group Sp(1). The convex hull of these 24 elements in 4-dimensional space form a convex regular 4-polytope called the 24-cell.
The binary tetrahedral group, denoted by 2T, fits into the short exact sequence
This sequence does not split, meaning that 2T is not a semidirect product of {±1} by T. In fact, there is no subgroup of 2T isomorphic to T.
The binary tetrahedral group is the covering group of the tetrahedral group. Thinking of the tetrahedral group as the alternating group on four letters, T ≅ A4, we thus have the binary tetrahedral group as the covering group, 2T ≅ .
The center of 2T is the subgroup {±1}. The inner automorphism group is isomorphic to A4, and the full automorphism group is isomorphic to S4.
The binary tetrahedral group can be written as a semidirect product
where Q is the quaternion group consisting of the 8 Lipschitz units and C3 is the cyclic group of order 3 generated by ω = −1/2(1 + i + j + k).
Official source
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