Part of the Atlas of Small Regular Polytopes

Polytope of Type {4,5}

Atlas Canonical Name {4,5}*160

▶ Play as a twisty puzzle

Overview

Group
SmallGroup(160,234)
Rank
3
Schläfli Type
{4,5}
Vertices, edges, …
16, 40, 20
Order of s0s1s2
5
Order of s0s1s2s1
4
Also known as
{4,5}5. if this polytope has another name.

Special Properties

  • Compact Hyperbolic Quotient
  • Locally Spherical
  • Non-Orientable

Quotients maximal quotients in bold

No regular quotients.

Covers minimal covers in bold

2-fold

3-fold

4-fold

5-fold

6-fold

7-fold

8-fold

9-fold

10-fold

11-fold

12-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*s0*s2*s1*s0*s1*s2*s1> of order 2

12 facets

8 vertex figures

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

10 facets

8 vertex figures

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

7 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

6 facets

4 vertex figures

Representations

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

References

None.

to this polytope.

Twisty Puzzle