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Graphics3D[GraphicsComplex[{{0, -1, (-3*Sqrt[3/2])/11}, {0, 1, (-3*Sqrt[3/2])/11},
   {-(1/Sqrt[3]), -1, (5*Sqrt[3/2])/22 + 1/(2*Sqrt[6])},
   {-(1/Sqrt[3]), 1, (5*Sqrt[3/2])/22 + 1/(2*Sqrt[6])}, {-1/(2*Sqrt[3]), -1/2,
    (5*Sqrt[3/2])/22 + 5/(2*Sqrt[6])}, {-1/(2*Sqrt[3]), 1/2, (5*Sqrt[3/2])/22 + 5/(2*Sqrt[6])},
   {1/(2*Sqrt[3]), -1/2, (5*Sqrt[3/2])/22 - 7/(2*Sqrt[6])},
   {1/Sqrt[3], 0, (5*Sqrt[3/2])/22 + 5/(2*Sqrt[6])},
   {2/Sqrt[3], 0, (5*Sqrt[3/2])/22 + 1/(2*Sqrt[6])}, {-Sqrt[3]/2, -1/2, (-3*Sqrt[3/2])/11},
   {-Sqrt[3]/2, 1/2, (-3*Sqrt[3/2])/11}, {Sqrt[3]/2, -1/2, (-3*Sqrt[3/2])/11},
   {Sqrt[3]/2, 1/2, (-3*Sqrt[3/2])/11}, {-(1/Sqrt[3]), 0, (5*Sqrt[3/2])/22 - 7/(2*Sqrt[6])},
   {1/(2*Sqrt[3]), 1/2, (5*Sqrt[3/2])/22 - 7/(2*Sqrt[6])}},
  Polygon[{{12, 13, 9}, {3, 10, 1}, {2, 11, 4}, {6, 5, 8}, {12, 9, 8, 5, 3, 1},
    {2, 4, 6, 8, 9, 13}, {10, 3, 5, 6, 4, 11}, {7, 14, 15}, {11, 2, 15, 14}, {1, 10, 14, 7},
    {13, 12, 7, 15}, {15, 2, 13}, {14, 10, 11}, {7, 12, 1}}]]]

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Johnson Polyhedra

Geometry