Part of the Atlas of Small Regular Polytopes

Polytope of Type {3,9}

Atlas Canonical Name {3,9}*648

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Overview

Group
SmallGroup(648,703)
Rank
3
Schläfli Type
{3,9}
Vertices, edges, …
36, 162, 108
Order of s0s1s2
12
Order of s0s1s2s1
9
Also known as
if this polytope has a name.

Special Properties

  • Compact Hyperbolic Quotient
  • Locally Spherical
  • Orientable

Quotients maximal quotients in bold

27-fold

Covers minimal covers in bold

2-fold

3-fold

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*(s2*s1*s0)^2*(s2*s1)^2> of order 3

36 facets

12 vertex figures

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

36 facets

12 vertex figures

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

36 facets

18 vertex figures

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

12 facets

6 vertex figures

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

12 facets

8 vertex figures

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

12 facets

4 vertex figures

Representations

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

References

None.

to this polytope.

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