Binary tetrahedral group

It follows that the binary tetrahedral group is a discrete subgroup of Spin(3) of order 24.

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.)

The group 2T has a presentation given by or equivalently, Generators with these relations are given by with

The quaternion group consisting of the 8 Lipschitz units forms a normal subgroup of 2T of index 3.

This group and the center {±1} are the only nontrivial normal subgroups.

All other subgroups of 2T are cyclic groups generated by the various elements, with orders 3, 4, and 6.

For all higher dimensions except A6 and A7 (corresponding to the 5-dimensional and 6-dimensional simplexes), this binary group is the covering group (maximal cover) and is superperfect, but for dimensional 5 and 6 there is an additional exceptional 3-fold cover, and the binary groups are not superperfect.

The binary tetrahedral group was used in the context of Yang–Mills theory in 1956 by Chen Ning Yang and others.

[5] It was first used in flavor physics model building by Paul Frampton and Thomas Kephart in 1994.

[6] In 2012 it was shown [7] that a relation between two neutrino mixing angles, derived [8] by using this binary tetrahedral flavor symmetry, agrees with experiment.