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Graphics3D[GraphicsComplex[{{0, -1, -1 - (13*(-27 + 5*Sqrt[6]))/772}, {0, -1, (-13*(-27 + 5*Sqrt[6]))/772},
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   {-1/(2*Sqrt[3]), -1/2, Sqrt[2/3] - (13*(-27 + 5*Sqrt[6]))/772},
   {-1/(2*Sqrt[3]), 1/2, Sqrt[2/3] - (13*(-27 + 5*Sqrt[6]))/772},
   {1/Sqrt[3], 0, Sqrt[2/3] - (13*(-27 + 5*Sqrt[6]))/772},
   {-Sqrt[3]/2, -1/2, -1 - (13*(-27 + 5*Sqrt[6]))/772}, {-Sqrt[3]/2, -1/2, (-13*(-27 + 5*Sqrt[6]))/772},
   {-Sqrt[3]/2, 1/2, -1 - (13*(-27 + 5*Sqrt[6]))/772}, {-Sqrt[3]/2, 1/2, (-13*(-27 + 5*Sqrt[6]))/772},
   {Sqrt[3]/2, -1/2, -1 - (13*(-27 + 5*Sqrt[6]))/772}, {Sqrt[3]/2, -1/2, (-13*(-27 + 5*Sqrt[6]))/772},
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    {14, 3, 4, 15}}]]]

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Johnson Polyhedra

Geometry