In mathematics, the **57-cell** (pentacontakaiheptachoron) is a self-dual abstract regular 4-polytope (four-dimensional polytope). Its 57 cells are hemi-dodecahedra. It also has 57 vertices, 171 edges and 171 two-dimensional faces. Its symmetry group is the projective special linear group L_{2}(19), so it has 3420 symmetries.

It has SchlĂ¤fli symbol {5,3,5} with 5 hemi-dodecahedral cells around each edge. It was discovered by H. S. M. Coxeter (1982).

The vertices and edges form the Perkel graph, the unique distance-regular graph with intersection array {6,5,2;1,1,3}, discovered by Manley Perkel (1979).