Quaternionic 288-Group

G_{288} = \frac{2 T _{L} \times 2 T _{R}}{{( 1 , 1 ) , ( - 1 , - 1 )}}

Element control

order 1

g^{0} / 1

Generator

[\frac{- 2}{2}, \frac{- 2}{2}]

Current

g^{0} = [\frac{- 2}{2}, \frac{- 2}{2}]

Quotient

(0, 0)

K_{32}

yes

Action

z \mapsto a z \overset{ˉ}{b}

full SO(4)

∣ G_{288} ∣ = 288

Current order: 1
Fixed-space dimension: 4
Characteristic polynomial: $= (x - 1)^{4}$
$= x^{4} - 4 x^{3} + 6 x^{2} - 4 x + 1$
Rotation angles: $(0, 0)$
Vertex displacementD11 → S1: $0$ 24 vertices
Left / Right tetrahedral action: identity/identity

basis $(1, i, j, k)$

M^{0}

1000010000100001

The tabs reveal different exact structures within the same 288 transformations of the projected 24-cell.