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== Symmetry==
[[File:Octagonal_tiling_with_444_mirror_lines.png|thumb|left|Octagonal tiling with *444 mirror lines, {{CDD|node_c1|split1-44|branch_c3-2|label4}}.]]
From [(4,4,4)] symmetry, there are 15 small index subgroups (7 unique) by mirror removal and alternation operators. 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. Adding 3 bisecting mirrors across each fundamental domains creates [[832 symmetry]]. The [[subgroup index]]-8 group, [(1<sup>+</sup>,4,1<sup>+</sup>,4,1<sup>+</sup>,4)] (222222) is the [[commutator subgroup]] of [(4,4,4)].
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