[1] The bidiakis cube is a cubic Hamiltonian graph and can be defined by the LCF notation [-6,4,-4]4.

The bidiakis cube can also be constructed from a cube by adding edges across the top and bottom faces which connect the centres of opposite sides of the faces.

With this construction, the bidiakis cube is a polyhedral graph, and can be realized as a convex polyhedron.

Therefore, by Steinitz's theorem, it is a 3-vertex-connected simple planar graph.

[2] The bidiakis cube is not a vertex-transitive graph and its full automorphism group is isomorphic to the dihedral group of order 8, the group of symmetries of a square, including both rotations and reflections.