Part of the Atlas of Small Regular Polytopes

Polytope of Type {8,8}

Atlas Canonical Name {8,8}*1296

▶ Play as a twisty puzzle

Overview

Group
SmallGroup(1296,3509)
Rank
3
Schläfli Type
{8,8}
Vertices, edges, …
81, 324, 81
Order of s0s1s2
6
Order of s0s1s2s1
4
Also known as
if this polytope has a name.

Special Properties

  • Compact Hyperbolic Quotient
  • Locally Spherical
  • Orientable
  • Self-Dual

Quotients maximal quotients in bold

No regular quotients.

Covers minimal covers in bold

None in this atlas.

Irregular Quotients of which this is a minimal cover

Click an entry to reveal its facets and vertex figures.

P/N, where N=<(s0*s1)^4*s2*s1*s0*s2*s1*s2> of order 3

27 facets

27 vertex figures

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

27 facets

27 vertex figures

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

27 facets

27 vertex figures

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

27 facets

27 vertex figures

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

27 facets

27 vertex figures

P/N, where N=<(s0*s1)^3*s2*(s1*s0)^2*s2*s1*s2, (s0*s1)^2*s2*(s1*s0)^2*s2*s1*s0*s2*s1> of order 9

9 facets

9 vertex figures

P/N, where N=<(s0*s1)^4*s2*s1*s0*s2*s1*s2, (s0*s1)^3*s0*s2*s1*s0*(s2*s1)^2> of order 9

9 facets

9 vertex figures

P/N, where N=<(s0*s1)^4*s2*s1*s0*s2*s1*s2, (s0*s1)^2*(s0*s2*s1)^2*s0*s1*s2*s1> of order 9

9 facets

9 vertex figures

Representations

Permutation Representation (GAP) 
s0 := ( 4,10)( 5,11)( 6,12)( 7,19)( 8,20)( 9,21)(16,22)(17,23)(18,24)(28,55)(29,56)(30,57)(31,64)(32,65)(33,66)(34,73)(35,74)(36,75)(37,58)(38,59)(39,60)(40,67)(41,68)(42,69)(43,76)(44,77)(45,78)(46,61)(47,62)(48,63)(49,70)(50,71)(51,72)(52,79)(53,80)(54,81);;
s1 := ( 2,10)( 3,19)( 4,55)( 5,64)( 6,73)( 7,28)( 8,37)( 9,46)(12,20)(13,56)(14,65)(15,74)(16,29)(17,38)(18,47)(22,57)(23,66)(24,75)(25,30)(26,39)(27,48)(31,61)(32,70)(33,79)(35,43)(36,52)(40,62)(41,71)(42,80)(45,53)(49,63)(50,72)(51,81)(59,67)(60,76)(69,77);;
s2 := ( 1, 2)( 4,20)( 5,19)( 6,21)( 7,11)( 8,10)( 9,12)(13,26)(14,25)(15,27)(16,17)(22,23)(28,29)(31,47)(32,46)(33,48)(34,38)(35,37)(36,39)(40,53)(41,52)(42,54)(43,44)(49,50)(55,56)(58,74)(59,73)(60,75)(61,65)(62,64)(63,66)(67,80)(68,79)(69,81)(70,71)(76,77);;
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*s0*s1*s0*s1*s0*s1, 
s0*s1*s0*s1*s2*s1*s2*s1*s0*s1*s0*s1*s2*s1*s2*s1, 
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1, 
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1, 
s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) 
s0 := Sym(81)!( 4,10)( 5,11)( 6,12)( 7,19)( 8,20)( 9,21)(16,22)(17,23)(18,24)(28,55)(29,56)(30,57)(31,64)(32,65)(33,66)(34,73)(35,74)(36,75)(37,58)(38,59)(39,60)(40,67)(41,68)(42,69)(43,76)(44,77)(45,78)(46,61)(47,62)(48,63)(49,70)(50,71)(51,72)(52,79)(53,80)(54,81);
s1 := Sym(81)!( 2,10)( 3,19)( 4,55)( 5,64)( 6,73)( 7,28)( 8,37)( 9,46)(12,20)(13,56)(14,65)(15,74)(16,29)(17,38)(18,47)(22,57)(23,66)(24,75)(25,30)(26,39)(27,48)(31,61)(32,70)(33,79)(35,43)(36,52)(40,62)(41,71)(42,80)(45,53)(49,63)(50,72)(51,81)(59,67)(60,76)(69,77);
s2 := Sym(81)!( 1, 2)( 4,20)( 5,19)( 6,21)( 7,11)( 8,10)( 9,12)(13,26)(14,25)(15,27)(16,17)(22,23)(28,29)(31,47)(32,46)(33,48)(34,38)(35,37)(36,39)(40,53)(41,52)(42,54)(43,44)(49,50)(55,56)(58,74)(59,73)(60,75)(61,65)(62,64)(63,66)(67,80)(68,79)(69,81)(70,71)(76,77);
poly := sub<Sym(81)|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*s0*s1*s0*s1*s0*s1, 
s0*s1*s0*s1*s2*s1*s2*s1*s0*s1*s0*s1*s2*s1*s2*s1, 
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1, 
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1, 
s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1 >;

References

None.

to this polytope.

Twisty Puzzle