The 57-cell, also called the pentacontaheptachoron, is a regular self-dual locally projective polytope with 57 hemidodecahedral facets
described by Coxeter (1982) and also constructed by Vanden Cruyce (1985; Hartley
and Leemans 2004). It has 57 vertices, 171 edges, 171 faces, and 57 cells (Coxeter
1982). It cannot be represented in 3-dimensional space in any reasonable way and
is highly self-intersecting even in 4-dimensional space because its boundary cells
are single-sided manifolds such as a Möbius strip
or Klein bottle (Séquin and Hamlin 2007).
Its symmetry group is the projective special linear group , of order 3420.

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