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I am curious about which properties of 'the simplest polyhedron' that represents the monster group symmetries are known.

I have of course noticed that there have been quite a few similar questions around. Like about how to construct the monster, or this very general question here and quite a few closely related questions like this on MO with very interesting information on the generators of the monster group.

However, I think I have more concrete questions on it, that I hope justify a separate question. I am interested in the/a simplest polyhedron representing of monster group symmetry.

My first question is:

  1. Is there a unique such object and does this have a name (e.g. like "icosahedron")?

I suppose there could be several possibilities (like the icosahedron and its dual the dodecahedron that represent the same point group symmetry $*532$ (in orbifold notation)).

Also, we know for example that it can be embedded in the 196882 dimensional Euklidian space. As it cannot be a simplex of that dimension (since then it should have the symmetry of the fully symmetric group) it must be a different polyhedron. My second question would be

  1. (Do we know) How many vertices does it have?

From the high symmetric polyhedra in 3D-Euclidian space we know that some are constructed/bound by (3-1) dimensional simplices (like the tetrahedron, octahedron, or the icosahedron) and some are not (like the cube and the dodecahedron). How about this 196882 dimensional "monster polyhedron"?

  1. Is the monster polyhedron constructed from $(196882-1=)196881$ dimensional simplices?

As it appears the group can be represented by 12 generators (and 80 relators) I ask myself

  1. Is it possible to give an explicit form for the cartesian coordinates of one generating vertex point (a vector with 196882 entries)?

That would depend as far as I understand on the explicit matrix form (definition and orientation) of the 12 generator elements. That would be 12 $196882\times196882$ matrices, which should be completely feasible with current computational resources (for example in quantum chemistry one diagonalizes such matrices in standard computations).

-Edit-

A very obvious question about its shape would also be:

  1. Are its facets (that is the $196882-1$-faces of the convex hull of its vertices) are $196882-1$-simplices?
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    $\begingroup$ Does that help? $\endgroup$
    – mrtaurho
    May 13 '20 at 7:18
  • $\begingroup$ @mrtaurho: Thank you! Yes, it answers 2. (97239461142009186000), gives a hint on 1. (probably no), and leaves still open 3. and 4. $\endgroup$ May 13 '20 at 11:00
  • $\begingroup$ I don't understand why you think the answer to 1. is "no": the permutation representation on the vertices of the polytope Conway mentions is the unique minimal permutation representation of the Monster, which implies that any polytope on which the Monster acts by symmetries has at least that many vertices, and any polytope with the same number of vertices gives the same permutation representation. By uniqueness did you not mean something along these lines, uniqueness of a minimal polytope such that the Monster is its symmetry group? $\endgroup$
    – Stephen
    May 14 '20 at 1:23
  • $\begingroup$ @Stephen: I understand that its unique, but I have not read the name, yet, except a technical description. You know something like "monsterhedron",... $\endgroup$ May 14 '20 at 6:36

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