Part of the Atlas of Small Regular Polytopes

Polytope of Type {12,6}

Atlas Canonical Name {12,6}*1296m

▶ Play as a twisty puzzle

Overview

Group
SmallGroup(1296,2909)
Rank
3
Schläfli Type
{12,6}
Vertices, edges, …
108, 324, 54
Order of s0s1s2
4
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

6-fold

9-fold

18-fold

81-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*s0*(s2*(s1*s0)^2)^2*s2*s1> of order 2

27 facets

54 vertex figures

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

27 facets

57 vertex figures

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

18 facets

36 vertex figures

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

36 facets

36 vertex figures

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

18 facets

36 vertex figures

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

12 facets

12 vertex figures

Representations

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

References

None.

to this polytope.

Twisty Puzzle