Acronym  ... 
Name  hyperbolic [(3^{3},4)^{4}] tiling 
Vertex figure  [(3^{3},4)^{4}] 
Confer 

Even so this tiling is uniform, it neither has a triangular fundamental domain, nor directly relates to those tilings with Coxeter domains.
This tiling could be obtained by a 2 step partial Stott contraction from o3o4x3*a: first color the triangles of that tiling in 4 colors, say yellow, brown, black, and green. Then apply a tripesic contraction, which would contract all yellow triangles into just vertices: this would result in paco3o4x3*a. Then apply a second such contraction onto that, reducing the brown triangles into points too. This would be the tiling under consideration. – One even could apply this again with respect to the black triangles. Then only the green triangles would be left, providing pexo3o4o3*a. (A fourth application then clearly would reduce the complete tiling into a single point.)
(N → ∞) N  4 4 4 4  6 6 4 [((3^3),4)^4] ++ 2  2N * * *  2 0 0 s:s 2  * 2N * *  0 2 0 g:g 2  * * 2N *  1 0 1 s:r 2  * * * 2N  0 1 1 g:r ++ 3  2 0 1 0  2N * * s {3} 3  0 2 0 1  * 2N * g {3} 4  0 0 2 2  * * N r {4}
pabexo3o4o3*a (ytriangle → pt & btriangle → pt) (N → ∞) N  2 2 2 2 4 4  6 6 4 [((3^3),4)^4] ++ 2  N * * * * *  2 0 0 s:y 2  * N * * * *  2 0 0 s:b 2  * * N * * *  0 2 0 g:y 2  * * * N * *  0 2 0 g:b 2  * * * * 2N *  1 0 1 s:r 2  * * * * * 2N  0 1 1 g:r ++ 3  1 1 0 0 1 0  2N * * s {3} 3  0 0 1 1 0 1  * 2N * g {3} 4  0 0 0 0 2 2  * * N r {4}
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