Simplest Canonical Polyhedra of Each Symmetry Type

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Simplest Canonical Polyhedron with S4 Symmetry
(1 of 2)
Vertices:  10  (4[3] + 2[4] + 4[5])
Faces:12  (4 isosceles triangles + {2 opposite * 2} obtuse triangles
    + {2 opposite * 2} irregular tetragons)
Edges:20  (4 different lengths)
Symmetry:  2-fold Rotoreflection  (S4)
Dual Solid:  Simplest S4 (2 of 2)
(values below based on edge-scribed radius = 1)
Edge 1 (4):  sqrt(root[(x^4)+12*(x^3)+48*(x^2)+464*x−400])    ≈0.88599887946423193324
Edge 2 (8):  sqrt(root[(x^4)+24*(x^3)+192*(x^2)−256*x−256])    ≈1.3099488129654297082
Edge 3 (4):  sqrt(root[(x^4)+28*(x^3)−112*(x^2)+80*x−16])    ≈1.6639837561142890394
Edge 4 (4):  sqrt(root[(x^4)+8*(x^3)−1024])    ≈2.0879336896154868143
Edge-scribed Radius:  1
Volume:sqrt(root[6561*(x^4)−1213056*(x^3)    
    +94224384*(x^2)+61341696*x−16122904576])    
≈3.73337236464520805853