Part of the Atlas of Small Regular Polytopes

Polytope of Type {12,6}

Atlas Canonical Name {12,6}*1296s

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Overview

Group
SmallGroup(1296,3528)
Rank
3
Schläfli Type
{12,6}
Vertices, edges, …
108, 324, 54
Order of s0s1s2
12
Order of s0s1s2s1
6
Also known as
if this polytope has a name.

Special Properties

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

Quotients maximal quotients in bold

3-fold

9-fold

18-fold

27-fold

36-fold

54-fold

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

27 facets

63 vertex figures

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

27 facets

54 vertex figures

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

30 facets

54 vertex figures

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

18 facets

36 vertex figures

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

18 facets

54 vertex figures

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

18 facets

36 vertex figures

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

18 facets

36 vertex figures

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

18 facets

36 vertex figures

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

18 facets

36 vertex figures

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

18 facets

36 vertex figures

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

18 facets

36 vertex figures

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

15 facets

36 vertex figures

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

12 facets

18 vertex figures

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

12 facets

18 vertex figures

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

9 facets

18 vertex figures

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

9 facets

21 vertex figures

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

9 facets

21 vertex figures

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

9 facets

27 vertex figures

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

6 facets

18 vertex figures

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

6 facets

24 vertex figures

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

6 facets

12 vertex figures

P/N, where N=<s0*s1*s2*s1*s0*s2, s1*s0*s1*s2*s1*s0*s2*s1> of order 9

6 facets

24 vertex figures

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

6 facets

12 vertex figures

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

6 facets

12 vertex figures

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

6 facets

12 vertex figures

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

6 facets

18 vertex figures

P/N, where N=<(s1*s2)^3, (s0*s1)^6> of order 12

6 facets

15 vertex figures

P/N, where N=<(s1*s2)^2, s0*(s1*s2)^2*s1*s0*s2> of order 18

3 facets

15 vertex figures

Representations

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

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