These 3 tilings are the flag-transitive ones. They should be very familiar to most people. All of them have various possible odd subsymmetries I won't list, as regulars tend to do. All three of these are regularly used in non-prismatic uniform honeycombs, but hexat and trat must adopt subsymmetry to do so.
1: Squat - Square tiling. Its faces are squares. Symbols are x4o4o, o4x4o, x4o4x, also s4o4o, o4s4o, s4o4s, and many diagrams with mixed s and x. The vertex figure is a square as well, as it is self-dual. It is also a comb product of two apeirogons, and can be considered as a stack of apeirogonal prisms. I will call square tiling symmetry "squattic".
2: Hexat - Hexagonal tiling. Its faces are hexagons. Symbols are x6o3o, x3x6o, x3x3x3*a. The vertex figure is a triangle. I will call hexagonal tiling symmetry "hexattic".
3: Trat - Triangular tiling. Its faces are triangles. Symbols are x3o6o, x3o3o3*a, also s6s3o and s3s3s3*a. The vertex figure is a hexagon. Can be considered as a stack of apeirogonal antiprisms, if you wanted to. It also has hexattic symmetry. Its regiment also contains a very important regular compound of 3 hexats, called fexat for fissal hexagonal tiling. Fexat will be used as a cell in many honeycombs.
Below shows how the regulars pair up as duals and conjugates.
Duals:
The following are self-dual: squat.
The following are dual pairs: hexat-trat.
Conjuates:
The following are self-conjugate: all (trivially).
The following are conjugate pairs: none.
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