Part of the Atlas of Small Regular Polytopes

Polytope of Type {4,4}

Atlas Canonical Name {4,4}*128

▶ Play as a twisty puzzle

Overview

Group
SmallGroup(128,928)
Rank
3
Schläfli Type
{4,4}
Vertices, edges, …
16, 32, 16
Order of s0s1s2
8
Order of s0s1s2s1
4
Also known as
{4,4}(4,0), {4,4|4}. 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

Covers minimal covers in bold

2-fold

3-fold

4-fold

5-fold

6-fold

7-fold

9-fold

10-fold

11-fold

13-fold

14-fold

15-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*s1)^2> of order 2

10 facets

8 vertex figures

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

8 facets

8 vertex figures

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

8 facets

8 vertex figures

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

8 facets

8 vertex figures

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

8 facets

10 vertex figures

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

5 facets

4 vertex figures

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

4 facets

4 vertex figures

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

4 facets

4 vertex figures

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

4 facets

4 vertex figures

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

6 facets

4 vertex figures

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

4 facets

6 vertex figures

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

4 facets

5 vertex figures

Representations

Permutation Representation (GAP) 
s0 := ( 5, 7)( 6, 8)(13,15)(14,16);;
s1 := ( 3, 4)( 7, 8)( 9,13)(10,14)(11,16)(12,15);;
s2 := ( 1, 9)( 2,10)( 3,11)( 4,12)( 5,13)( 6,14)( 7,15)( 8,16);;
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, s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) 
s0 := Sym(16)!( 5, 7)( 6, 8)(13,15)(14,16);
s1 := Sym(16)!( 3, 4)( 7, 8)( 9,13)(10,14)(11,16)(12,15);
s2 := Sym(16)!( 1, 9)( 2,10)( 3,11)( 4,12)( 5,13)( 6,14)( 7,15)( 8,16);
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, s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1 >;

References

None.

to this polytope.

Twisty Puzzle