The snub cubical graph is the Archimedean graph on 24 nodes and 60 edges obtained by taking the skeleton of the snub cube. It is a quintic graph, is planar, Hamiltonian, and has chromatic number 3. Several embeddings are illustrated above.
It is implemented in the Wolfram Language as GraphData["SnubCubicalGraph"].
It is vertex-transitive, although not edge-transitive because some edges are part of three-circuits while others are not.
The snub cubical graph has 
distinct (directed) Hamiltonian cycles, giving two LCF-type
notations of order 4 (illustrated above), 12 of order 3 (illustrated above), 627
of order 2, and 70127 of order 1.
Its graph spectrum is given by 
,
where 
,
, and 
are the roots of 
. Its automorphism
group is of order 
.
