Truncated tetrahexagonal tiling

The dual of the tiling represents the fundamental domains of (*642) orbifold symmetry.

From [6,4] symmetry, there are 15 small index subgroup by mirror removal and alternation operators.

In these images unique mirrors are colored red, green, and blue, and alternately colored triangles show the location of gyration points.

The [6+,4+], (32×) subgroup has narrow lines representing glide reflections.

Larger subgroup constructed as [6,4*], removing the gyration points of [6,4+], (3*22), index 6 becomes (*3333), and [6*,4], removing the gyration points of [6+,4], (2*33), index 12 as (*222222).

Truncated tetrahexagonal tiling with mirror lines in green, red, and blue:
Symmetry diagrams for small index subgroups of [6,4], shown in a hexagonal translational cell within a {6,6} tiling , with a fundamental domain in yellow.