Viewed in perspective outside of a Poincaré disk model, this shows one hexagonal tiling cell within the honeycomb, and its mid-radius horosphere (the horosphere incident with edge midpoints). In this projection, the hexagons grow infinitely small towards the infinite boundary asymptoting towards a single ideal point. It can be seen as similar to the order-3 apeirogonal tiling, {∞,3} of H^{2}, with horocycle circumscribing vertices of apeirogonal faces.

It has a total of five reflectional constructions from five related Coxeter groups all with four mirrors and only the first being regular: [6,3,3], [3,6,3], [6,3,6], [6,3^{[3]}] and [3^{[3,3]}] , having 1, 4, 6, 12 and 24 times larger fundamental domains respectively. In Coxeter notation subgroup markups, they are related as: [6,(3,3)^{*}] (remove 3 mirrors, index 24 subgroup); [3,6,3^{*}] or [3^{*},6,3] (remove 2 mirrors, index 6 subgroup); [1^{+},6,3,6,1^{+}] (remove two orthogonal mirrors, index 4 subgroup); all of these are isomorphic to [3^{[3,3]}]. The ringed Coxeter diagrams are , , , and , representing different types (colors) of hexagonal tilings in the Wythoff construction.

It is one of 15 regular hyperbolic honeycombs in 3-space, 11 of which like this one are paracompact, with infinite cells or vertex figures.

It is one of 15 uniform paracompact honeycombs in the [6,3,3] Coxeter group, along with its dual, the order-6 tetrahedral honeycomb, {3,3,6}.

It is in a sequence with regular polychora: 5-cell {3,3,3}, tesseract {4,3,3}, 120-cell {5,3,3} of Euclidean 4-space, with tetrahedral vertex figures.

It is a part of sequence of regular honeycombs of the form {6,3,p}, with hexagonal tiling cells:

The **rectified hexagonal tiling honeycomb**, t_{1}{6,3,3}, has tetrahedral and trihexagonal tiling facets, with a triangular prism vertex figure. The half-symmetry construction alternate two types of tetrahedra.

The **truncated hexagonal tiling honeycomb**, t_{0,1}{6,3,3}, has tetrahedral and truncated hexagonal tiling facets, with a tetrahedral vertex figure.

It is similar to the 2D hyperbolic truncated order-3 apeirogonal tiling, t{∞,3} with apeirogonal and triangle faces:

The **bitruncated hexagonal tiling honeycomb** or **bitruncated order-6 tetrahedral honeycomb**, t_{1,2}{6,3,3}, has truncated tetrahedra and hexagonal tiling cells, with a tetrahedral vertex figure.

The **cantellated hexagonal tiling honeycomb**, t_{0,2}{6,3,3}, has octahedral and rhombitrihexagonal tiling cells, with a triangular prism vertex figure.

The **cantitruncated hexagonal tiling honeycomb**, t_{0,1,2}{6,3,3}, has truncated tetrahedron and truncated trihexagonal tiling cells, with a tetrahedron vertex figure.

The **runcinated hexagonal tiling honeycomb**, t_{0,3}{6,3,3}, has tetrahedron, rhombitrihexagonal tiling hexagonal prism, triangular prism cells, with a octahedron vertex figure.

The **runcitruncated hexagonal tiling honeycomb**, t_{0,1,3}{6,3,3}, has cuboctahedron, Triangular prism, Dodecagonal prism, and truncated hexagonal tiling cells, with a quad-pyramid vertex figure.

The **runcicantellated hexagonal tiling honeycomb** or **runcitruncated order-6 tetrahedral honeycomb**, t_{0,2,3}{6,3,3}, has truncated tetrahedron, hexagonal prism, hexagonal prism, and rhombitrihexagonal tiling cells, with a quad-pyramid vertex figure.

The **omnitruncated hexagonal tiling honeycomb** or **omnitruncated order-6 tetrahedral honeycomb**, t_{0,1,2,3}{6,3,3}, has truncated octahedron, hexagonal prism, dodecagonal prism, and truncated trihexagonal tiling cells, with a quad-pyramid vertex figure.