Part of the Atlas of Small Regular Polytopes

Polytope of Type {3,3,3,3}

Atlas Canonical Name {3,3,3,3}*720

Overview

Group
SmallGroup(720,763)
Rank
5
Schläfli Type
{3,3,3,3}
Vertices, edges, …
6, 15, 20, 15, 6
Order of s0s1s2s3s4
6
Order of s0s1s2s3s4s3s2s1
3
Also known as
5-simplex, {3,3,3,3}. if this polytope has another name.

Special Properties

  • Universal
  • Spherical
  • Locally Spherical
  • Orientable
  • Self-Dual

Quotients maximal quotients in bold

No regular quotients.

Covers minimal covers in bold

2-fold

Irregular Quotients of which this is a minimal cover

None.

Representations

Permutation Representation (GAP) 
s0 := (4,6);;
s1 := (5,6);;
s2 := (3,5);;
s3 := (2,3);;
s4 := (1,2);;
poly := Group([s0,s1,s2,s3,s4]);;
Finitely Presented Group Representation (GAP) 
F := FreeGroup("s0","s1","s2","s3","s4");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  s4 := F.5;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s4*s4, s0*s2*s0*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s0*s4*s0*s4, 
s1*s4*s1*s4, s2*s4*s2*s4, s0*s1*s0*s1*s0*s1, 
s1*s2*s1*s2*s1*s2, s2*s3*s2*s3*s2*s3, 
s3*s4*s3*s4*s3*s4 ];;
poly := F / rels;;
Permutation Representation (Magma) 
s0 := Sym(6)!(4,6);
s1 := Sym(6)!(5,6);
s2 := Sym(6)!(3,5);
s3 := Sym(6)!(2,3);
s4 := Sym(6)!(1,2);
poly := sub<Sym(6)|s0,s1,s2,s3,s4>;
Finitely Presented Group Representation (Magma) 
poly<s0,s1,s2,s3,s4> := Group< s0,s1,s2,s3,s4 | s0*s0, s1*s1, s2*s2, 
s3*s3, s4*s4, s0*s2*s0*s2, s0*s3*s0*s3, 
s1*s3*s1*s3, s0*s4*s0*s4, s1*s4*s1*s4, 
s2*s4*s2*s4, s0*s1*s0*s1*s0*s1, s1*s2*s1*s2*s1*s2, 
s2*s3*s2*s3*s2*s3, s3*s4*s3*s4*s3*s4 >;

References

  1. Schläfli, L.; Theorie Der Vielfachen Kontinuität, Denkschriften Der Schweizerischen Naturforschenden Gesellschaft, 38, pp1–237 (1901)

to this polytope.