Part of the Atlas of Small Regular Polytopes

Polytope of Type {4,4}

Atlas Canonical Name {4,4}*256

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Overview

Group
SmallGroup(256,6661)
Rank
3
Schläfli Type
{4,4}
Vertices, edges, …
32, 64, 32
Order of s0s1s2
8
Order of s0s1s2s1
8
Also known as
{4,4}(4,4), {4,4}8. if this polytope has another name.

Special Properties

  • Toroidal
  • Locally Spherical
  • Orientable
  • Self-Dual

Quotients maximal quotients in bold

2-fold

4-fold

8-fold

16-fold

32-fold

Covers minimal covers in bold

2-fold

3-fold

5-fold

7-fold

Irregular Quotients of which this is a minimal cover

Click an entry to reveal its facets and vertex figures.

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

16 facets

16 vertex figures

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

18 facets

16 vertex figures

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

16 facets

16 vertex figures

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

16 facets

18 vertex figures

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

8 facets

10 vertex figures

P/N, where N=<(s0*s1)^2, s0*s1*(s2*s1*s0)^3*s2> of order 4

10 facets

8 vertex figures

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

8 facets

8 vertex figures

P/N, where N=<s1*s0*s2*s1*s0*s1*s2*s1, s0*s1*s0*s2*s1*s0*s1*s2*s1*s0*s2*s1*s2> of order 4

9 facets

8 vertex figures

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

8 facets

9 vertex figures

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

8 facets

8 vertex figures

P/N, where N=<(s0*s1)^2, s2*s1*s0*s1*s2*s1*s0*s2*s1*s2> of order 8

5 facets

5 vertex figures

Representations

Permutation Representation (GAP) 
s0 := ( 1, 9)( 2,10)( 3,11)( 4,12)( 5,13)( 6,14)( 7,15)( 8,16);;
s1 := ( 5, 7)( 6, 8)(11,12)(13,14)(15,16);;
s2 := ( 1, 5)( 2, 6)( 3, 8)( 4, 7)( 9,13)(10,14)(11,16)(12,15);;
poly := Group([s0,s1,s2]);;
Finitely Presented Group Representation (GAP) 
F := FreeGroup("s0","s1","s2");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  
rels := [ s0*s0, s1*s1, s2*s2, s0*s2*s0*s2, s0*s1*s0*s1*s0*s1*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2, s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) 
s0 := Sym(16)!( 1, 9)( 2,10)( 3,11)( 4,12)( 5,13)( 6,14)( 7,15)( 8,16);
s1 := Sym(16)!( 5, 7)( 6, 8)(11,12)(13,14)(15,16);
s2 := Sym(16)!( 1, 5)( 2, 6)( 3, 8)( 4, 7)( 9,13)(10,14)(11,16)(12,15);
poly := sub<Sym(16)|s0,s1,s2>;
Finitely Presented Group Representation (Magma) 
poly<s0,s1,s2> := Group< s0,s1,s2 | s0*s0, s1*s1, s2*s2, 
s0*s2*s0*s2, s0*s1*s0*s1*s0*s1*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2, s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1 >;

References

None.

to this polytope.

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