Symmetry
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∞).
| Small index subgroups of [∞,4], (*∞42) |
|---|
| Index |
1 |
2 |
4 |
| Diagram |
 |
 |
 |
 |
 |
 |
| Coxeter |
[∞,4]
     |
[1+,∞,4]
 =    |
[∞,4,1+]
=    |
[∞,1+,4]
=   |
[1+,∞,4,1+]
= |
[∞+,4+]
  |
| Orbifold |
*∞42 |
*∞44 |
*∞∞2 |
*∞222 |
*∞2∞2 |
∞2× |
| Semidirect subgroups |
| Diagram |
|
 |
 |
 |
 |
 |
| Coxeter |
|
[∞,4+]
|
[∞+,4]
|
[(∞,4,2+)]
  |
[1+,∞,1+,4]
= =
= = |
[∞,1+,4,1+]
= =
= = |
| Orbifold |
|
4*∞ |
∞*2 |
2*∞2 |
∞*22 |
2*∞∞ |
| Direct subgroups |
| Index |
2 |
4 |
8 |
| Diagram |
 |
 |
 |
 |
 |
| Coxeter |
[∞,4]+
= |
[∞,4+]+
= |
[∞+,4]+
= |
[∞,1+,4]+
  = |
[∞+,4+]+ = [1+,∞,1+,4,1+]
 = = = |
| Orbifold |
∞42 |
∞44 |
∞∞2 |
∞222 |
∞2∞2 |
| Radical subgroups |
| Index |
|
8 |
∞ |
16 |
∞ |
| Diagram |
|
 |
 |
 |
 |
| Coxeter |
|
[∞,4*]
  =  |
[∞*,4]
  |
[∞,4*]+
= |
[∞*,4]+
|
| Orbifold |
|
*∞∞∞∞ |
*2∞ |
∞∞∞∞ |
2∞ |
