The regular complex polytope, 3{6}2 or or , represents the Cayley diagram for the binary tetrahedral group, with each red and blue triangle a directed subgraph.[1]
In mathematics, the binary tetrahedral group, denoted 2T or ⟨2,3,3⟩,[2] 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 preimage 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.)
^Coxeter, Complex Regular Polytopes, p 109, Fig 11.5E
^ Coxeter&Moser: Generators and Relations for discrete groups: <l,m,n>: Rl = Sm = Tn = RST
and 22 Related for: Binary tetrahedral group information
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