Heptagonal-truncated tetrahedral duoprism

{{Infobox polytope The heptagonal-truncated tetrahedral duoprism or hetut is a convex uniform duoprism that consists of 7 truncated tetrahedral prisms, 4 hexagonal-heptagonal duoprisms and 4 triangular-heptagonal duoprisms. Each vertex joins 2 truncated tetrahedral prisms, 1 triangular-heptagonal duoprism, and 2 hexagonal-heptagonal duoprisms.
 * type=Uniform
 * dim = 5
 * img=
 * off = auto
 * obsa = Hetut
 * coxeter = x7o x3x3o
 * army = Hetut
 * reg = Hetut
 * terons = 4 triangular-heptagonal duoprisms, 7 truncated tetrahedral prisms, 4 hexagonal-heptagonal duoprisms
 * cells = 28 triangular prisms, 28 hexagonal prisms, 7 truncated tetrahedra, 6+12 heptagonal prisms
 * faces = 28 triangles, 42+84 squares, 28 hexagons, 12 heptagons
 * edges = 42+84+84
 * vertices = 84
 * circum = $$\sqrt{\frac{{11}{8}+\frac{1}{4\sin^2\frac\pi7}} ≈ 1.64408$$
 * hypervol = $$\frac{161\sqrt2}{48\tan\frac\pi7} ≈ 9.85000$$
 * dit = Tuttip–tut–tuttip: $$\frac{5\pi}{7} ≈ 128.57143°$$
 * dit2 = Theddip-hep-haheddip: $$\arccos\left(-\frac13\right) ≈ 109.47122°$$
 * dit3 = Theddip–trip–tut: 90°
 * dit4 = Haheddip-hip-tut: 90°
 * dit5 = Haheddip–hep–haheddip: $$\arccos\left(\frac13\right) ≈ 70.52877°$$
 * verf = Isosceles triangular scalene, edge lengths 1, $\sqrt{3}$, $\sqrt{3}$ (base triangle), 2cos(π/7) (top), $\sqrt{2}$ (side edges)
 * symmetry = A{{sub|3}}×I2(7), order 336
 * pieces = 15
 * loc = 30
 * dual=Heptagonal-triakis tetrahedral duotegum
 * conjugate = Heptagrammic-truncated tetrahedral duoprism, Great heptagrammic-truncated tetrahedral duoprism
 * conv = Yes
 * orientable=Yes
 * nat=Tame}}

Vertex coordinates
The vertices of a heptagonal-truncated tetrahedral duoprism of edge length 2sin(π/7) are given by all permutations and even sign changes of the last three coordinates of: where j = 2, 4, 6.
 * $$\left(1,\,0,\,\frac{3\sqrt2\sin\frac\pi7}2,\,\frac{\sqrt2\sin\frac\pi7}2,\,\frac{\sqrt2\sin\frac\pi7}2\right),$$
 * $$\left(\cos\left(\frac{j\pi}{7}\right),\,±\sin\left(\frac{j\pi}{7}\right),\,\frac{3\sqrt2\sin\frac\pi7}2,\,\frac{\sqrt2\sin\frac\pi7}2,\,\frac{\sqrt2\sin\frac\pi7}2\right),$$