Part of the Atlas of Small Regular Polytopes

Polytope of Type {4,8}

Atlas Canonical Name {4,8}*576a

▶ Play as a twisty puzzle

Overview

Group
SmallGroup(576,5339)
Rank
3
Schläfli Type
{4,8}
Vertices, edges, …
36, 144, 72
Order of s0s1s2
24
Order of s0s1s2s1
6
Also known as
if this polytope has a name.

Special Properties

  • Compact Hyperbolic Quotient
  • Locally Spherical
  • Orientable

Quotients maximal quotients in bold

2-fold

4-fold

8-fold

9-fold

18-fold

36-fold

72-fold

Covers minimal covers in bold

2-fold

3-fold

Irregular Quotients of which this is a minimal cover

Click an entry to reveal its facets and vertex figures.

P/N, where N=<(s1*s0*s1*s2)^3> of order 2

36 facets

18 vertex figures

P/N, where N=<(s0*s1)^2*(s2*s1*s0*s1)^2*s2> of order 2

36 facets

18 vertex figures

P/N, where N=<(s0*s1)^2> of order 2

40 facets

18 vertex figures

P/N, where N=<(s0*s1)^2*(s2*s1*s0)^2*(s1*s2)^2> of order 3

24 facets

12 vertex figures

P/N, where N=<(s1*s0*s1*s2)^2> of order 3

24 facets

12 vertex figures

P/N, where N=<(s1*s0*s1*s2)^2, (s0*s1*s2*s1)^3> of order 6

12 facets

6 vertex figures

P/N, where N=<(s0*s2*s1)^2*s0*s1*s2*s1> of order 6

12 facets

6 vertex figures

P/N, where N=<(s1*s0*s1*s2)^2, s0*s1*s0*(s2*s1*s0*s1)^2*s2*s1> of order 6

12 facets

6 vertex figures

P/N, where N=<(s0*s1)^2, s0*s2*s1*s0*s1*s2> of order 6

16 facets

6 vertex figures

P/N, where N=<(s0*s1)^2, (s2*s1*s0)^2*(s1*s2)^2> of order 6

16 facets

6 vertex figures

Representations

Permutation Representation (GAP) 
s0 := ( 4, 7)( 5, 8)( 6, 9)(13,16)(14,17)(15,18)(22,25)(23,26)(24,27)(31,34)(32,35)(33,36)(40,43)(41,44)(42,45)(49,52)(50,53)(51,54)(58,61)(59,62)(60,63)(67,70)(68,71)(69,72);;
s1 := ( 2, 4)( 3, 7)( 6, 8)(11,13)(12,16)(15,17)(19,28)(20,31)(21,34)(22,29)(23,32)(24,35)(25,30)(26,33)(27,36)(37,55)(38,58)(39,61)(40,56)(41,59)(42,62)(43,57)(44,60)(45,63)(46,64)(47,67)(48,70)(49,65)(50,68)(51,71)(52,66)(53,69)(54,72);;
s2 := ( 1,38)( 2,37)( 3,39)( 4,41)( 5,40)( 6,42)( 7,44)( 8,43)( 9,45)(10,47)(11,46)(12,48)(13,50)(14,49)(15,51)(16,53)(17,52)(18,54)(19,65)(20,64)(21,66)(22,68)(23,67)(24,69)(25,71)(26,70)(27,72)(28,56)(29,55)(30,57)(31,59)(32,58)(33,60)(34,62)(35,61)(36,63);;
poly := Group([s0,s1,s2]);;
Finitely Presented Group Representation (GAP) 
F := FreeGroup("s0","s1","s2");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  
rels := [ s0*s0, s1*s1, s2*s2, s0*s2*s0*s2, s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) 
s0 := Sym(72)!( 4, 7)( 5, 8)( 6, 9)(13,16)(14,17)(15,18)(22,25)(23,26)(24,27)(31,34)(32,35)(33,36)(40,43)(41,44)(42,45)(49,52)(50,53)(51,54)(58,61)(59,62)(60,63)(67,70)(68,71)(69,72);
s1 := Sym(72)!( 2, 4)( 3, 7)( 6, 8)(11,13)(12,16)(15,17)(19,28)(20,31)(21,34)(22,29)(23,32)(24,35)(25,30)(26,33)(27,36)(37,55)(38,58)(39,61)(40,56)(41,59)(42,62)(43,57)(44,60)(45,63)(46,64)(47,67)(48,70)(49,65)(50,68)(51,71)(52,66)(53,69)(54,72);
s2 := Sym(72)!( 1,38)( 2,37)( 3,39)( 4,41)( 5,40)( 6,42)( 7,44)( 8,43)( 9,45)(10,47)(11,46)(12,48)(13,50)(14,49)(15,51)(16,53)(17,52)(18,54)(19,65)(20,64)(21,66)(22,68)(23,67)(24,69)(25,71)(26,70)(27,72)(28,56)(29,55)(30,57)(31,59)(32,58)(33,60)(34,62)(35,61)(36,63);
poly := sub<Sym(72)|s0,s1,s2>;
Finitely Presented Group Representation (Magma) 
poly<s0,s1,s2> := Group< s0,s1,s2 | s0*s0, s1*s1, s2*s2, 
s0*s2*s0*s2, s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1 >;

References

None.

to this polytope.

Twisty Puzzle