Part of the Atlas of Small Regular Polytopes

Polytope of Type {4,12}

Atlas Canonical Name {4,12}*1296

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Overview

Group
SmallGroup(1296,2909)
Rank
3
Schläfli Type
{4,12}
Vertices, edges, …
54, 324, 162
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

6-fold

9-fold

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

81 facets

27 vertex figures

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

81 facets

27 vertex figures

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

90 facets

30 vertex figures

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

54 facets

18 vertex figures

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

54 facets

36 vertex figures

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

54 facets

18 vertex figures

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

45 facets

15 vertex figures

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

45 facets

18 vertex figures

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

36 facets

12 vertex figures

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

36 facets

12 vertex figures

P/N, where N=<(s0*s1)^2, s0*(s1*s2)^3*s1*s0*s2> of order 6

30 facets

20 vertex figures

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

18 facets

12 vertex figures

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

12 facets

8 vertex figures

Representations

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

References

None.

to this polytope.

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