Pentacontahexahecatonicosihexaexon

The pentacontahexahecatonicosihexaexon, or lin, also called the 132 polytope, is a convex uniform polyexon. It has 56 pentacontatetrapeta and 126 demihexeracts as facets, with 7 of each at a vertex forming a birectified heptapeton as the vertex figure.

The pentacontahexahecatonicosihexaexon contains the vertices of a birectified hexacontatetrapetal prism, a cuboctahedral duoprismatic prism, a cubic-icositetrachoric duoprism, a hepteract, a small cellated octaexon, a triangular-small prismated hexateric duoprism, and a hexagonal-dodecateric duoprism.

It can tile 7-dimensional Euclidean space by itself, forming the pentacontahexahecatonicosihexaexic heptacomb.

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

Representations
A pentacontahexahecatonicosihexaexon has the following Coxeter diagrams:


 * o3o3o3x *c3o3o3o (full symmetry)
 * oooo3oxoo3oooo3ooxo3oooo *c3xoox&#xt (E6 axial, pentacontatetrapeton-first)
 * xooox3ooooo3ooxoo *b3oxoxo3ooooo3ooxoo&#xt (D6 axial, demihexeract-first)