Part of the Atlas of Small Regular Polytopes

Polytope of Type {3,4,3}

Atlas Canonical Name {3,4,3}*1152

Overview

Group
SmallGroup(1152,157478)
Rank
4
Schläfli Type
{3,4,3}
Vertices, edges, …
24, 96, 96, 24
Order of s0s1s2s3
12
Order of s0s1s2s3s2s1
4
Also known as
24-cell, {3,4,3}. if this polytope has another name.

Special Properties

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

Quotients maximal quotients in bold

2-fold

32-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*s0*s2)^2*s1> of order 2

14 facets

14 vertex figures

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

12 facets

13 vertex figures

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

13 facets

12 vertex figures

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

8 facets

8 vertex figures

P/N, where N=<s0*(s1*s0*s2)^2*s1, (s1*s2)^2*(s3*s2*s1*s0*s2*s1*s2)^2*s3> of order 4

8 facets

7 vertex figures

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

9 facets

9 vertex figures

P/N, where N=<s0*(s1*s0*s2)^2*s1, s0*(s2*s3*s2*s1*s0*s2*s1)^2*s2*s3> of order 4

7 facets

8 vertex figures

P/N, where N=<s0*(s1*s0*s2)^2*s1, s0*s2*s1*s0*s3*s2*s1*s0*s2*s1*s3*s2> of order 4

9 facets

9 vertex figures

Representations

Permutation Representation (GAP) 
s0 := ( 5, 8)( 6, 7)(11,13)(12,14)(19,24)(20,23);;
s1 := ( 3, 8)( 4, 7)(13,16)(14,15)(19,21)(20,22);;
s2 := ( 3, 4)( 9,22)(10,21)(11,20)(12,19)(13,23)(14,24)(15,18)(16,17);;
s3 := ( 1,10)( 2, 9)( 3,16)( 4,15)( 5,11)( 6,12)( 7,14)( 8,13)(17,18);;
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, s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;
Permutation Representation (Magma) 
s0 := Sym(24)!( 5, 8)( 6, 7)(11,13)(12,14)(19,24)(20,23);
s1 := Sym(24)!( 3, 8)( 4, 7)(13,16)(14,15)(19,21)(20,22);
s2 := Sym(24)!( 3, 4)( 9,22)(10,21)(11,20)(12,19)(13,23)(14,24)(15,18)(16,17);
s3 := Sym(24)!( 1,10)( 2, 9)( 3,16)( 4,15)( 5,11)( 6,12)( 7,14)( 8,13)(17,18);
poly := sub<Sym(24)|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, 
s1*s2*s1*s2*s1*s2*s1*s2 >;

References

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

to this polytope.