Part of the Atlas of Small Regular Polytopes

Polytope of Type {3,3,4}

Atlas Canonical Name {3,3,4}*384

Overview

Group
SmallGroup(384,5602)
Rank
4
Schläfli Type
{3,3,4}
Vertices, edges, …
8, 24, 32, 16
Order of s0s1s2s3
8
Order of s0s1s2s3s2s1
4
Also known as
4-cross-polytope, {3,3,4}. if this polytope has another name.

Special Properties

  • Universal
  • Spherical
  • Locally Spherical
  • Orientable

Quotients maximal quotients in bold

2-fold

8-fold

Covers minimal covers in bold

2-fold

Irregular Quotients of which this is a minimal cover

Click an entry to reveal its facets and vertex figures.

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

8 facets

6 vertex figures

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

8 facets

5 vertex figures

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

4 facets

5 vertex figures

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

4 facets

4 vertex figures

Representations

Permutation Representation (GAP) 
s0 := (3,5)(4,6);;
s1 := (5,7)(6,8);;
s2 := (1,7)(2,8);;
s3 := (3,4)(5,6)(7,8);;
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, 
s1*s2*s1*s2*s1*s2, s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;
Permutation Representation (Magma) 
s0 := Sym(8)!(3,5)(4,6);
s1 := Sym(8)!(5,7)(6,8);
s2 := Sym(8)!(1,7)(2,8);
s3 := Sym(8)!(3,4)(5,6)(7,8);
poly := sub<Sym(8)|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, s1*s2*s1*s2*s1*s2, 
s2*s3*s2*s3*s2*s3*s2*s3 >;

References

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

to this polytope.