Pentacontahexahecatonicosihexapentacosiheptacontahexaexon

The pentacontahexahecatonicosihexapentacosiheptacontahexaexon, or lanq, also called the rectified pentacontahexahecatonicosahexaexon, rolin, or rectified 132 polytope, is a convex uniform polyexon. It has 56 rectified pentacontatetrapeta, 126 birectified hexeracts, and 576 birectified heptapeta. 4 rectified pentacontatetrapeta, 3 birectified hexeracts, and 2 birectified heptapeta join at each triangular-tetrahedral duoprismatic prismatic vertex. It is the rectified pentacontahexahecatonicosihexaexon, the birectified pentacontahexapentacosiheptacontahexaexon, and the trirectified hecatonicosihexapentacosiheptacontahexaexon. It can therefore be considered in some ways to be a hybrid of the 3 main polytopes of E7 symmetry.

A Pentacontahexahecatonicosihexapentacosiheptacontahexaexon contains the vertices and edges of a peticellirhombated hexadecaexon, a cellirhombated demihexeractic prism, and a hexagonal-celliprismatotruncated dodecateral duoprism.

Vertex coordinates
The vertices of a pentacontahexahecatonicosihexapentacosiheptacontahexaexon of edge length 1, centered at the origin, are given by:


 * $$\left(±\frac{\sqrt2}{2},\,±\frac{\sqrt2}{2},\,±\frac{\sqrt2}{2},\,±\frac{\sqrt2}{2},\,0,\,0,\,±2\right)$$ and all permutations of first 6 coordinates,
 * $$\left(\frac{3\sqrt2}{4},\,\frac{3\sqrt2}{4},\,\frac{3\sqrt2}{4},\,\frac{\sqrt2}{4},\,\frac{\sqrt2}{4},\,\frac{\sqrt2}{4},\,±\frac32\right)$$ and all permutations and even sign changes of the first 6 coordinates,
 * $$\left(±\sqrt2,\,±\sqrt2,\,±\frac{\sqrt2}{2},\,±\frac{\sqrt2}{2},\,0,\,0,\,±1\right)$$ and all permutations of first 6 coordinates,
 * $$\left(\frac{5\sqrt2}{4},\,\frac{3\sqrt2}{4},\,\frac{3\sqrt2}{4},\,\frac{\sqrt2}{4},\,\frac{\sqrt2}{4},\,\frac{\sqrt2}{4},\,±\frac12\right)$$ and all permutations and odd sign changes of the first 6 coordinates,
 * $$\left(±\frac{3\sqrt2}{2},\,±\frac{\sqrt2}{2},\,±\frac{\sqrt2}{2},\,±\frac{\sqrt2}{2},\,0,\,0,\,0\right)$$ and all permutations of first 6 coordinates,
 * $$\left(\sqrt2,\,\sqrt2,\,\frac{\sqrt2}{2},\,\frac{\sqrt2}{2},\,\frac{\sqrt2}{2},\,\frac{\sqrt2}{2},\,0\right)$$ and all permutations and odd sign changes of the first 6 coordinates,
 * $$\left(±\sqrt2,\,±\sqrt2,\,±\sqrt2,\,0,\,0,\,0,\,0\right)$$ and all permutations of first 6 coordinates.

Representations
A pentacontahexahecatonicosihexapentacosiheptacontahexaexon has the following Coxeter diagrams:


 * o3o3x3o *c3o3o3o (full symmetry)
 * ooxoooo3oxoxooo3xoxoxox3oooxoxo3ooooxoo *c3oxoooxo&#xt (E6 axial, rectified pentacontatetrapeton-first)
 * oxoo(ooo)ooxo3xoxo(xoo)oxox3ooox(ouo)xooo *b3oxox(oou)xoxo3ooxo(oxo)oxoo3ooox(uoo)xooo&#xt (D6 axial, birectified hexeract-first)