Truncated tetraapeirogonal tiling

In geometry, the truncated tetraapeirogonal tiling is a semiregular tiling of the hyperbolic plane. There are one square, one octagon, and one apeirogon on each vertex. It has Schläfli symbol of tr{∞,4}. The dual of this tiling represents the fundamental domains of [∞,4], (∞42) symmetry. There are 15 small index subgroups constructed from [∞,4] by mirror removal and alternation. Mirrors can be removed if its branch orders are all even, and cuts neighboring branch orders in half. Removing two mirrors leaves a half-order gyration point where the removed mirrors met. In these images fundamental domains are alternately colored black and white, and mirrors exist on the boundaries between colors. The subgroup index-8 group, [1+,∞,1+,4,1+] (∞2∞2) is the commutator subgroup of [∞,4]. A larger subgroup is constructed as [∞,4], index 8, as [∞,4+], (4∞) with gyration points removed, becomes (∞∞∞∞) or (∞4), and another [∞,4], index ∞ as [∞+,4], (∞2) with gyration points removed as (2∞). And their direct subgroups [∞,4]+, [∞,4]+, subgroup indices 16 and ∞ respectively, can be given in orbifold notation as (∞∞∞∞) and (2∞).