Part of the Atlas of Small Regular Polytopes

Polytope of Type {6,9}

Atlas Canonical Name {6,9}*1296c

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Overview

Group
SmallGroup(1296,1788)
Rank
3
Schläfli Type
{6,9}
Vertices, edges, …
72, 324, 108
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

Quotients maximal quotients in bold

3-fold

4-fold

9-fold

12-fold

27-fold

36-fold

54-fold

108-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)^3> of order 2

60 facets

36 vertex figures

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

54 facets

36 vertex figures

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

36 facets

24 vertex figures

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

36 facets

24 vertex figures

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

42 facets

24 vertex figures

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

27 facets

18 vertex figures

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

27 facets

18 vertex figures

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

18 facets

12 vertex figures

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

18 facets

12 vertex figures

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

24 facets

12 vertex figures

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

24 facets

12 vertex figures

P/N, where N=<(s1*s0)^2*s1*s2*s1*s0*s2*s1*s2, (s0*s1)^2*(s2*s1*s0)^2*s2*s1> of order 12

9 facets

6 vertex figures

Representations

Permutation Representation (GAP) 
s0 := ( 3, 4)( 5, 9)( 6,10)( 7,12)( 8,11)(15,16)(17,21)(18,22)(19,24)(20,23)(27,28)(29,33)(30,34)(31,36)(32,35);;
s1 := ( 2, 4)( 5, 9)( 6,12)( 7,11)( 8,10)(13,29)(14,32)(15,31)(16,30)(17,25)(18,28)(19,27)(20,26)(21,33)(22,36)(23,35)(24,34);;
s2 := ( 1,14)( 2,13)( 3,15)( 4,16)( 5,22)( 6,21)( 7,23)( 8,24)( 9,18)(10,17)(11,19)(12,20)(25,26)(29,34)(30,33)(31,35)(32,36);;
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*s0*s1*s0*s1, 
s2*s0*s1*s2*s1*s2*s1*s2*s0*s1*s2*s1*s2*s1, 
s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1, 
s0*s1*s0*s2*s1*s0*s1*s2*s1*s0*s1*s0*s2*s1*s0*s2*s1*s0*s1*s2*s1*s0*s1*s0*s2*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) 
s0 := Sym(36)!( 3, 4)( 5, 9)( 6,10)( 7,12)( 8,11)(15,16)(17,21)(18,22)(19,24)(20,23)(27,28)(29,33)(30,34)(31,36)(32,35);
s1 := Sym(36)!( 2, 4)( 5, 9)( 6,12)( 7,11)( 8,10)(13,29)(14,32)(15,31)(16,30)(17,25)(18,28)(19,27)(20,26)(21,33)(22,36)(23,35)(24,34);
s2 := Sym(36)!( 1,14)( 2,13)( 3,15)( 4,16)( 5,22)( 6,21)( 7,23)( 8,24)( 9,18)(10,17)(11,19)(12,20)(25,26)(29,34)(30,33)(31,35)(32,36);
poly := sub<Sym(36)|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*s0*s1*s0*s1, 
s2*s0*s1*s2*s1*s2*s1*s2*s0*s1*s2*s1*s2*s1, 
s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1, 
s0*s1*s0*s2*s1*s0*s1*s2*s1*s0*s1*s0*s2*s1*s0*s2*s1*s0*s1*s2*s1*s0*s1*s0*s2*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1 >;

References

None.

to this polytope.

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