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Graphics3D[GraphicsComplex[{{-1, (-52*Sqrt[2] - Sqrt[3])/940, -1/2}, {-1, (-52*Sqrt[2] - Sqrt[3])/940, 1/2},
{-1/2, -Sqrt[3]/2 + (-52*Sqrt[2] - Sqrt[3])/940, -1/2}, {-1/2, -Sqrt[3]/2 + (-52*Sqrt[2] - Sqrt[3])/940,
1/2}, {-1/2, Sqrt[3]/2 + (-52*Sqrt[2] - Sqrt[3])/940, -1/2},
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{1/2, Sqrt[3]/2 + (-52*Sqrt[2] - Sqrt[3])/940, -1/2}, {1/2, Sqrt[3]/2 + (-52*Sqrt[2] - Sqrt[3])/940, 1/2},
{1, (-52*Sqrt[2] - Sqrt[3])/940, -1/2}, {1, (-52*Sqrt[2] - Sqrt[3])/940, 1/2},
{(-3 - Sqrt[6])/4, (-52*Sqrt[2] - Sqrt[3])/940 + Root[1 - 160*#1^2 + 256*#1^4 & , 4, 0], 0},
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Polygon[{{7, 3, 1, 5, 9, 11}, {12, 10, 6, 2, 4, 8}, {9, 5, 6, 10}, {1, 3, 4, 2}, {3, 7, 8, 4},
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{13, 6, 5}}]]]
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Johnson Polyhedra
Geometry