Are there any finite regular abstract 5-polytopes whose facets are 11-cells and whose vertex figures are 57-cells?
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I surely don't know what structure it would live in and I neither do know whether the below factor $n$ allows for any finite value at all, but your amalgamation problem simply returns the following incidence matrix:
$$\begin{array}{r|r|r|r|r} 22n & 57 & 171 & 171 & 57\\ \hline 2 & 627n & 6 & 15 & 10\\ \hline 3 & 3 & 1254n & 5 & 5\\ \hline 6 & 15 & 10 & 627n & 2\\ \hline 11 & 55 & 55 & 11 & 114n \end{array}$$
--- rk
