Geometry of the Binary Tetrahedral Group in SU(2)

Vertices

The elements of order six are +/-a, +/-a^2, +/-b, +/-b^2, +/-c, +/-c^2, +/-d, +/-d^2, where

a = (1/2) ( 1- i- j- k),
b = (1/2) ( 1- i+ j+ k),
c = (1/2) ( 1+ i- j+ k),
d = (1/2) ( 1+ i+ j- k).

The elements of order four are +/- i, +/- j, +/- k;
1 and - 1 complete the list of 24 elements.

(See The Quaternionic Representation of the Binary Tetrahedral Group for the derivation of these elements from symmetries of the tetrahedron.)

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