Part of the Atlas of Small Regular Polytopes

Polytope of Type {12,12}

Atlas Canonical Name {12,12}*1296h

▶ Play as a twisty puzzle

Overview

Group
SmallGroup(1296,3529)
Rank
3
Schläfli Type
{12,12}
Vertices, edges, …
54, 324, 54
Order of s0s1s2
6
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

3-fold

9-fold

18-fold

36-fold

54-fold

108-fold

162-fold

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*s2*s1)^3> of order 2

27 facets

27 vertex figures

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

27 facets

27 vertex figures

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

30 facets

30 vertex figures

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

18 facets

18 vertex figures

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

18 facets

18 vertex figures

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

18 facets

18 vertex figures

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

18 facets

18 vertex figures

P/N, where N=<(s0*s1)^6, s2*(s1*s0)^2*s2*s1*s0*(s2*s1)^2*s2> of order 4

15 facets

15 vertex figures

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

12 facets

12 vertex figures

Representations

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

References

None.

to this polytope.

Twisty Puzzle