Part of the Atlas of Small Regular Polytopes

Polytope of Type {6,3,3}

Atlas Canonical Name {6,3,3}*1296

Overview

Group
SmallGroup(1296,3490)
Rank
4
Schläfli Type
{6,3,3}
Vertices, edges, …
54, 108, 54, 12
Order of s0s1s2s3
12
Order of s0s1s2s3s2s1
6
Also known as
3T4(3,0), {{6,3}6,{3,3}}. if this polytope has another name.

Special Properties

  • Universal
  • Locally Toroidal
  • Orientable

Quotients maximal quotients in bold

27-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=<(s1*s0*s1*s2)^2> of order 3

8 facets

18 vertex figures

P/N, where N=<s1*s0*s1*s2*s1*s0*s3*s2*(s1*s0)^2*s2*s1*s3*s2> of order 3

6 facets

18 vertex figures

P/N, where N=<(s0*s1)^2*s2*s1*s3*s2*(s1*s0)^2*s2*s1*s3*s2> of order 3

4 facets

18 vertex figures

P/N, where N=<(s1*s0*s1*s2)^2, s1*s0*s1*s3*s2*s1*s0*s1*s2*s3> of order 9

6 facets

6 vertex figures

P/N, where N=<(s1*s0*s1*s2)^2, (s0*s1)^2*s0*s3*s2*s1*s0*s1*s2*s3> of order 9

4 facets

6 vertex figures

P/N, where N=<s0*s3*s2*s1*s0*s1*s2*s3, (s1*s0*s1*s2)^2> of order 9

4 facets

  • 4 of 3-fold non-regular quotient of {6,3}*108

6 vertex figures

Representations

Permutation Representation (GAP) 
s0 := ( 2, 3)( 5, 6)( 8, 9)(11,12);;
s1 := (1,9)(2,8)(3,7);;
s2 := ( 7,10)( 8,11)( 9,12);;
s3 := ( 4,10)( 5,12)( 6,11);;
poly := Group([s0,s1,s2,s3]);;
Finitely Presented Group Representation (GAP) 
F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s2*s0*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s1*s2*s1*s2*s1*s2, 
s2*s3*s2*s3*s2*s3, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s1*s2*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) 
s0 := Sym(12)!( 2, 3)( 5, 6)( 8, 9)(11,12);
s1 := Sym(12)!(1,9)(2,8)(3,7);
s2 := Sym(12)!( 7,10)( 8,11)( 9,12);
s3 := Sym(12)!( 4,10)( 5,12)( 6,11);
poly := sub<Sym(12)|s0,s1,s2,s3>;
Finitely Presented Group Representation (Magma) 
poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2, 
s3*s3, s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s1*s2*s1*s2*s1*s2, s2*s3*s2*s3*s2*s3, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s1*s2*s1 >;

References

  1. Theorem 11B5, McMullen P., Schulte, E.; Abstract Regular Polytopes (Cambridge University Press, 2002)

to this polytope.