57-cell

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.

The symmetry order is 3420, from the product of the number of cells (57) and the symmetry of each cell (60). The symmetry abstract structure is the projective special linear group of the 2-dimensional vector space over the finite field of 19 elements, L2(19).

It has Schläfli type {5,3,5} with 5 hemi-dodecahedral cells around each edge. It was discovered by.

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