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).
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).
See also
11-cell – abstract regular polytope with hemi-icosahedral cells.
120-cell – regular 4-polytope with dodecahedral cells
Order-5 dodecahedral honeycomb - regular hyperbolic honeycomb with same Schläfli type, {5,3,5}. (The 57-cell can be considered as being derived from it by identification of appropriate elements.)
