Hecatonicosihexapentacosiheptacontahexaexon

The hecatonicosihexapentacosiheptacontahexaexon, or naq, also called the 321 polytope or biambohexadecaexon, is a convex uniform polyexon. It has 126 hexacontatetrapeta and 576 heptapeta as facets, with 27 hexacontatetrapeta and 72 heptapeta at a vertex forming an icosiheptaheptacontadipeton as the vertex figure.

The hecatonicosihexapentacosiheptacontahexaexon contains the vertices of a hexacontatetrapetic prism and a triangular-hexateric duoprism. It is also the convex hull of two oppositely oriented rectified octaexa.

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

Alternate coordinates can be given in 8 dimensions, as all permutations of: These give a naq of edge length $$4\sqrt2$$, and come from the fact that naq is the convex hull of two opposite.
 * $$±\left(3,\,3,\,-1,\,-1,\,-1,\,-1,\,-1,\,-1\right)$$.

Representations
A hecatonicosihexapentacosiheptacontahexaexon has the following Coxeter diagrams:


 * o3o3o3o *c3o3o3x (full symmetry)
 * oxo3ooo3ooo *b3ooo3ooo3xox&#xt (D6 axial, hexacontatetrapeton-first)
 * oxoo3oooo3oooo3oooo3ooxo *c3oooo&#xt (E6 axial, vertex-first)
 * xoxox oxooo3ooooo3oooxo *c3ooooo3ooxoo&#xt (D5×A1 axial, edge-first)
 * xooo3ooxo3oooo3oooo3oxoo3ooox&#xt (A6 axial, heptapeton-first)
 * oo3xo3oo3oo3oo3ox3oo&#zx (A7 symmetry, hull of 2 rectified octaexa)
 * ox xo3oo3oo *c3oo3oo3ox&#zx (D6×A1 symmetry)
 * xxoo xoxo xoox oxoo3oooo3ooxo *e3ooox&#zx (D4×A1×A1×A1 symmetry)