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Graphics3D[GraphicsComplex[{{-1, 0, Sqrt[-1 + Sqrt[3]]/2}, {-1/2, -Sqrt[3]/2, Sqrt[-1 + Sqrt[3]]/2},
   {-1/2, Sqrt[3]/2, Sqrt[-1 + Sqrt[3]]/2}, {-1/2, -1/(2*Sqrt[3]),
    Sqrt[5/3 + Sqrt[3] + 4*Sqrt[(2*(-1 + Sqrt[3]))/3]]/2}, {0, -1, -Sqrt[-1 + Sqrt[3]]/2},
   {0, 1, -Sqrt[-1 + Sqrt[3]]/2}, {0, 1/Sqrt[3], Sqrt[5/3 + Sqrt[3] + 4*Sqrt[(2*(-1 + Sqrt[3]))/3]]/2},
   {1/2, -1/(2*Sqrt[3]), Sqrt[5/3 + Sqrt[3] + 4*Sqrt[(2*(-1 + Sqrt[3]))/3]]/2},
   {1/2, -Sqrt[3]/2, Sqrt[-1 + Sqrt[3]]/2}, {1/2, Sqrt[3]/2, Sqrt[-1 + Sqrt[3]]/2},
   {1, 0, Sqrt[-1 + Sqrt[3]]/2}, {-(1/Sqrt[3]), 0, -Sqrt[5/3 + Sqrt[3] + 4*Sqrt[(2*(-1 + Sqrt[3]))/3]]/2},
   {1/(2*Sqrt[3]), -1/2, -Sqrt[5/3 + Sqrt[3] + 4*Sqrt[(2*(-1 + Sqrt[3]))/3]]/2},
   {1/(2*Sqrt[3]), 1/2, -Sqrt[5/3 + Sqrt[3] + 4*Sqrt[(2*(-1 + Sqrt[3]))/3]]/2},
   {-Sqrt[3]/2, -1/2, -Sqrt[-1 + Sqrt[3]]/2}, {-Sqrt[3]/2, 1/2, -Sqrt[-1 + Sqrt[3]]/2},
   {Sqrt[3]/2, -1/2, -Sqrt[-1 + Sqrt[3]]/2}, {Sqrt[3]/2, 1/2, -Sqrt[-1 + Sqrt[3]]/2}},
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    {16, 15, 1}, {15, 5, 2}, {5, 17, 9}, {17, 18, 11}, {7, 4, 8}, {4, 7, 3, 1}, {8, 4, 2, 9}, {7, 8, 11, 10},
    {10, 3, 7}, {1, 2, 4}, {9, 11, 8}, {12, 14, 13}, {18, 17, 13, 14}, {16, 6, 14, 12}, {5, 15, 12, 13},
    {13, 17, 5}, {14, 6, 18}, {12, 15, 16}}]]]

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Johnson Polyhedra

Geometry