This tiling represents a hyperbolic kaleidoscope of 6 mirrors defining a regular hexagon fundamental domain.
In Coxeter notation can be represented as [6*,4], removing two of three mirrors (passing through the hexagon center).
Adding a bisecting mirror through 2 vertices of a hexagonal fundamental domain defines a trapezohedral *4422 symmetry.
They are similar to 7 of the uniform colorings of the square tiling, but exclude 2 cases with order-2 gyrational symmetry.
This tiling is also topologically related as a part of sequence of regular polyhedra and tilings with four faces per vertex, starting with the octahedron, with Schläfli symbol {n,4}, and Coxeter diagram , with n progressing to infinity.