Heptagonal-cuboctahedral duoprism

{{Infobox polytope The heptagonal-cuboctahedral duoprism or heco is a convex uniform duoprism that consists of 7 cuboctahedral prisms, 6 square-heptagonal duoprisms, and 8 triangular-heptagonal duoprisms. Each vertex joins 2 cuboctahedral prisms, 2 triangular-heptagonal duoprisms, and 2 square-heptagonal duoprisms.
 * type=Uniform
 * dim = 5
 * img=
 * off = auto
 * obsa = Heco
 * coxeter = x7o o4x3o
 * army = Heco
 * reg = Heco
 * terons = 7 cuboctahedral prisms, 8 triangular-heptagonal duoprisms, 6 square-heptagonal duoprisms
 * cells = 56 triangular prisms, 42 cubes, 7 cuboctahedra, 24 heptagonal prisms
 * faces = 56 triangles, 42+168 squares, 12 heptagons
 * edges = 84+168
 * vertices = 84
 * circum = $$\frac{\sqrt{4+\frac{1}{\sin^2\frac\pi7}}{2} ≈ 1.52577$$
 * hypervol = $$\frac{35\sqrt2}{12\tan\frac\pi7} ≈ 8.56521$$
 * dit = Cope–co–cope: $$\frac{5\pi}{7} ≈ 128.57143°$$
 * dit2 = Theddip–hep–squahedip: $$\arccos\left(-\frac{\sqrt3}{3}\right) ≈ 125.26439°$$
 * dit3 = Theddip–trip–cope: 90°
 * dit4 = Squahedip–cube–cope: 90°
 * verf = Rectangular scalene, edge lengths 1, $\sqrt{2}$, 1, $\sqrt{2}$ (base rectangle), 2cos(π/7) (top), $\sqrt{2}$ (side edges)
 * symmetry = BC{{sub|3}}×I2(7), order 672
 * pieces = 21
 * loc = 20
 * dual=Heptagonal-rhombic dodecahedral duotegum
 * conjugate = Heptagrammic-cuboctahedral duoprism, Great heptagrammic-cuboctahedral duoprism
 * conv = Yes
 * orientable=Yes
 * nat=Tame}}

Vertex coordinates
The vertices of a heptagonal-cuboctahedral duoprism of edge length 2sin(π/7) are given by all permutations of the last three coordinates of: where j = 2, 4, 6.
 * $$\left(1,\,0,\,0,\,±\sqrt2\sin\frac\pi7,\,±\sqrt2\sin\frac\pi7\right),$$
 * $$\left(\cos\left(\frac{j\pi}7\right),\,±\sin\left(\frac{j\pi}7\right),\,0,\,±\sqrt2\sin\frac\pi7,\,±\sqrt2\sin\frac\pi7\right),$$

Representations
A heptagonal-cuboctahedral duoprism has the following Coxeter diagrams:
 * x7o o4x3o (full symmetry)
 * x7o x3o3x