Part of the Atlas of Small Regular Polytopes

Polytope of Type {3,6,3}

Atlas Canonical Name {3,6,3}*240

Overview

Group
SmallGroup(240,189)
Rank
4
Schläfli Type
{3,6,3}
Vertices, edges, …
5, 20, 20, 5
Order of s0s1s2s3
5
Order of s0s1s2s3s2s1
6
Also known as
7T4(2,0)(2,0), {{3,6}4,{6,3}4}. if this polytope has another name.

Special Properties

  • Universal
  • Locally Toroidal
  • Orientable
  • Self-Dual

Quotients maximal quotients in bold

2-fold

Covers minimal covers in bold

2-fold

3-fold

4-fold

5-fold

6-fold

7-fold

8-fold

Irregular Quotients of which this is a minimal cover

None.

Representations

Permutation Representation (GAP) 
s0 := (4,5);;
s1 := (3,4);;
s2 := (2,3)(6,7);;
s3 := (1,2)(6,7);;
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, s0*s1*s0*s1*s0*s1, 
s2*s3*s2*s3*s2*s3, s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1, 
s1*s2*s1*s3*s2*s1*s3*s2*s1*s2*s3*s2 ];;
poly := F / rels;;
Permutation Representation (Magma) 
s0 := Sym(7)!(4,5);
s1 := Sym(7)!(3,4);
s2 := Sym(7)!(2,3)(6,7);
s3 := Sym(7)!(1,2)(6,7);
poly := sub<Sym(7)|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, 
s0*s1*s0*s1*s0*s1, s2*s3*s2*s3*s2*s3, 
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1, 
s1*s2*s1*s3*s2*s1*s3*s2*s1*s2*s3*s2 >;

References

  1. Theorem 11E10, McMullen P., Schulte, E.; Abstract Regular Polytopes (Cambridge University Press, 2002)

to this polytope.