Simplest Canonical Polyhedra of Each Symmetry Type

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Simplest Canonical Polyhedron with S6 Symmetry
(1 of 2)
Vertices:  14  (6[3] + 6[5] + 2[6])
Faces:18  (6 isosceles triangles + {3 opposite * 2} obtuse triangles
    + {3 opposite * 2} irregular tetragons)
Edges:30  (4 different lengths)
Symmetry:  3-fold Rotoreflection  (S6)
Dual Solid:  Simplest S6 (2 of 2)
(values below based on edge-scribed radius = 1)
Edge 1 (12):  sqrt(root[3*(x^4)+128*(x^3)+1376*(x^2)−512*x−256])    ≈0.79370114654735575787
Edge 2 (6):  sqrt(root[3*(x^4)+8*(x^3)+16*(x^2)+128*x−256])    ≈1.2027426572743596596
Edge 3 (6):  sqrt(root[(x^4)+64*(x^3)+896*(x^2)+4096*x−12288])    ≈1.4135461293691891078
Edge 4 (6):  sqrt(root[3*(x^4)+200*(x^3)−944*(x^2)+896*x−256])    ≈1.82258764009619300953
Edge-scribed Radius:  1
Volume:sqrt(root[3*(x^4)+416*(x^3)+139520*(x^2)    
    +12525568*x−243859456])    
≈4.0416513574567349260