Hexagonal antitegum

The hexagonal antitegum or hate, also known as the hexagonal trapezohedron, is an antitegum based on the hexagon, constructed as the dual of a hexagonal antiprism. It has 12 kites as faces, with 2 order 6 and 12 order 3 vertices. .

Each face of this polyhedron is a kite with its longer edges $$2+\sqrt3 ≈ 3.73205$$ times the length of its shorter edges.