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