Simplest Canonical Polyhedra of Each Symmetry Type

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Simplest Canonical Polyhedron with S6 Symmetry
(2 of 2)
Vertices:  18  (6[3] + 6[3] + 6[4])
Faces:14  ({3 opposite * 2} acute triangles
    + {3 opposite * 2} irregular pentagons + 2 triambi)
Edges:30  (6 short + 12 medium + 12 long)
Symmetry:  3-fold Rotoreflection  (S6)
Dual Solid:  Simplest S6 (1 of 2)
(values below based on edge-scribed radius = 1)
Short Edge (6):  root[77*(x^4)+128*(x^3)−20*(x^2)−64*x+16]    ≈0.503637283942245852849
Medium Edge (12):  root[11*(x^4)−32*(x^3)+40*(x^2)−16]    ≈0.88898210064483747367
Long Edge (12):  root[7*(x^4)−24*(x^3)+80*(x^2)−128*x+64]    ≈0.93447391882979791242
Edge-scribed Radius:  1
Volume:sqrt(root[4322241*(x^4)−1036014624*(x^3)    
    +141962413824*(x^2)−1105988526080*x    
    −2129398530048])    
≈3.16072041524436595647