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Graphics3D[GraphicsComplex[{{0, -(1/Sqrt[2]), 1/Sqrt[2] + (167 - 44*Sqrt[2])/376},
   {0, 1/Sqrt[2], 1/Sqrt[2] + (167 - 44*Sqrt[2])/376}, {-(1/Sqrt[2]), 0, 1/Sqrt[2] + (167 - 44*Sqrt[2])/376},
   {1/Sqrt[2], 0, 1/Sqrt[2] + (167 - 44*Sqrt[2])/376}, {1/2, (1 + Sqrt[2])/2, -1 + (167 - 44*Sqrt[2])/376},
   {1/2, (1 + Sqrt[2])/2, (167 - 44*Sqrt[2])/376}, {-1/2, (1 + Sqrt[2])/2, -1 + (167 - 44*Sqrt[2])/376},
   {-1/2, (1 + Sqrt[2])/2, (167 - 44*Sqrt[2])/376}, {(-1 - Sqrt[2])/2, 1/2, -1 + (167 - 44*Sqrt[2])/376},
   {(-1 - Sqrt[2])/2, 1/2, (167 - 44*Sqrt[2])/376}, {(-1 - Sqrt[2])/2, -1/2, -1 + (167 - 44*Sqrt[2])/376},
   {(-1 - Sqrt[2])/2, -1/2, (167 - 44*Sqrt[2])/376}, {-1/2, (-1 - Sqrt[2])/2, -1 + (167 - 44*Sqrt[2])/376},
   {-1/2, (-1 - Sqrt[2])/2, (167 - 44*Sqrt[2])/376}, {1/2, (-1 - Sqrt[2])/2, -1 + (167 - 44*Sqrt[2])/376},
   {1/2, (-1 - Sqrt[2])/2, (167 - 44*Sqrt[2])/376}, {(1 + Sqrt[2])/2, -1/2, -1 + (167 - 44*Sqrt[2])/376},
   {(1 + Sqrt[2])/2, -1/2, (167 - 44*Sqrt[2])/376}, {(1 + Sqrt[2])/2, 1/2, -1 + (167 - 44*Sqrt[2])/376},
   {(1 + Sqrt[2])/2, 1/2, (167 - 44*Sqrt[2])/376}},
  Polygon[{{3, 2, 8, 10}, {1, 3, 12, 14}, {4, 1, 16, 18}, {2, 4, 20, 6}, {6, 8, 2}, {10, 12, 3}, {14, 16, 1},
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    {11, 13, 14, 12}, {13, 15, 16, 14}, {15, 17, 18, 16}, {17, 19, 20, 18}, {19, 5, 6, 20}}]]]

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Johnson Polyhedra