The Klein map is the regular map of type 
on an orientable surface
of genus 3, where the subscript 8 gives the length of a
Petrie polygon. It has 56 vertices,
84 edges, and 24 heptagonal faces.
Its automorphism group has order 336, with
an orientation-preserving subgroup of order 168 (Schulte and Wills 2012).
The dual map is the Klein map 
, with 24 vertices, 84
edges, and 56 triangular faces.
The dual graph is the 24-Klein graph, while the skeleton of the 
map is the 56-Klein graph, also known as the Foster
census graph 
.
McCooey (2009) constructed a non-self-intersecting integer-coordinate polyhedron realizing the 
map with 24 planar nonconvex heptagonal faces. Its geometric
symmetry group is the chiral tetrahedral group, a proper subgroup of the full combinatorial
automorphism group.
(Basic Shape)."