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Graphics3D[GraphicsComplex[{{0, 0, (-39 - 4*Sqrt[6])/75 + (3 + Sqrt[6])/3},
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   {-1/(2*Sqrt[3]), 1/2, (-39 - 4*Sqrt[6])/75}, {-1/(2*Sqrt[3]), 1/2, 1 + (-39 - 4*Sqrt[6])/75},
   {1/Sqrt[3], 0, (-39 - 4*Sqrt[6])/75}, {1/Sqrt[3], 0, 1 + (-39 - 4*Sqrt[6])/75}},
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Johnson Polyhedra

Geometry