Part of the Atlas of Small Regular Polytopes

Polytope of Type {6,12}

Atlas Canonical Name {6,12}*1296n

▶ Play as a twisty puzzle

Overview

Group
SmallGroup(1296,2909)
Rank
3
Schläfli Type
{6,12}
Vertices, edges, …
54, 324, 108
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*(s2*s1)^2*s0*s2*s1> of order 2

54 facets

27 vertex figures

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

57 facets

27 vertex figures

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

36 facets

18 vertex figures

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

36 facets

36 vertex figures

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

36 facets

18 vertex figures

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

12 facets

12 vertex figures

Representations

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

References

None.

to this polytope.

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