Birectified hecatonicosihexapentacosiheptacontahexaexon

The birectified hecatonicosihexapentacosiheptacontahexaexon, or branq, also called the birectified 321 polytope, is a convex uniform polyexon. It has 56 rectified icosiheptaheptacontadipeta, 126 birectified hexacontatetrapeta, and 576 birectified heptapeta. 3 rectified icosiheptaheptacontadipeta, 5 birectified hexacontatetrapeta, and 5 birectified heptapeta join at each triangular-rectified pentachoric duoprismatic vertex. As the name suggests, it is the birectification of the hecatonicosihexapentacosiheptacontahexaexon.

A birectified hecatonicosihexapentacosiheptacontahexaexon contains the vertices and edges of a small prismated hexacontatetrapetal prism, a terirhombated octaexon, a triangular-prismatotruncated hexateral duoprism, a hexagonal-cellirhombated dodecateral duoprism, a cubic-rectified icositetrachoric duoprism, and a cuboctahedral-truncated octahedral duoprismatic prism.

Vertex coordinates
The vertices of a birectified hecatonicosihexapentacosiheptacontahexaexon of edge length 1, centered at the origin, are given by all permutations of first 6 coordinates of all permutations and even sign changes of the first 6 coordinates of and all permutations and odd sign changes of the first 6 coordinates of
 * $$\left(±\frac{\sqrt2}{2},\,±\frac{\sqrt2}{2},\,±\frac{\sqrt2}{2},\,0,\,0,\,0,\,±\frac32\right),$$
 * $$\left(±\sqrt2,\,±\frac{\sqrt2}{2},\,±\frac{\sqrt2}{2},\,±\frac{\sqrt2}{2},\,0,\,0,\,±\frac12\right),$$
 * $$\left(\frac{3\sqrt2}{4},\,\frac{3\sqrt2}{4},\,\frac{\sqrt2}{4},\,\frac{\sqrt2}{4},\,\frac{\sqrt2}{4},\,\frac{\sqrt2}{4},\,±1\right),$$
 * $$\left(\frac{5\sqrt2}{4},\,\frac{\sqrt2}{4},\,\frac{\sqrt2}{4},\,\frac{\sqrt2}{4},\,\frac{\sqrt2}{4},\,\frac{\sqrt2}{4},\,0\right),$$
 * $$\left(\frac{3\sqrt2}{4},\,\frac{3\sqrt2}{4},\,\frac{3\sqrt2}{4},\,\frac{\sqrt2}{4},\,\frac{\sqrt2}{4},\,\frac{\sqrt2}{4},\,0\right).$$

Representations
A birectified hecatonicosihexapentacosiheptacontahexaexon has the following Coxeter diagrams:


 * o3o3o3o *c3x3o3o (full symmetry)
 * oooxoo3xoxooo3oxooxo3oooxox3ooxooo *c3oooooo&#xt (E6 axial, rectified icosiheptaheptacontadipeton-first)
 * oxo(xo)oxo3oox(oo)xoo3ooo(ox)ooo *b3xoo(ox)oox3oxo(oo)oxo3oox(uo)xoo&#xt (D6 axial, birectified hexacontatetrapeton-first)