Properties
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, we thus have the binary tetrahedral group as the covering group,
The center of 2T is the subgroup {±1}. The outer automorphism group is trivial, so that the inner automorphism group is isomorphic to the full automorphism group, which is the tetrahedral group T.
The binary tetrahedral group can be written as a semidirect product
where Q is the quaternion group consisting of the 8 Lipschitz units and Z3 is the cyclic group of order 3 generated by ω = −½(1+i+j+k). The group Z3 acts on the normal subgroup Q by conjugation. Conjugation by ω is the automorphism of Q that cyclically rotates i, j, and k.
One can show that the binary tetrahedral group is isomorphic to the special linear group SL(2,3) — the group of all 2×2 matrices over the finite field F3 with unit determinant, with this isomorphism covering the isomorphism of the projective special linear group PSL(2,3) with the alternating group
Read more about this topic: Binary Tetrahedral Group
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