Part of the Atlas of Small Regular Polytopes

Polytope of Type {6,5}

Atlas Canonical Name {6,5}*960

▶ Play as a twisty puzzle

Overview

Group
SmallGroup(960,11358)
Rank
3
Schläfli Type
{6,5}
Vertices, edges, …
96, 240, 80
Order of s0s1s2
5
Order of s0s1s2s1
5
Also known as
if this polytope has a name.

Special Properties

  • Compact Hyperbolic Quotient
  • Locally Spherical
  • Non-Orientable

Quotients maximal quotients in bold

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

40 facets

48 vertex figures

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

44 facets

48 vertex figures

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

28 facets

32 vertex figures

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

26 facets

24 vertex figures

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

22 facets

24 vertex figures

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

24 facets

24 vertex figures

P/N, where N=<(s0*s1)^3, s0*s2*(s1*s0)^2*s1*s2, s1*s2*s1*s0*s2*(s1*s0)^2*(s1*s2)^2*s1> of order 8

16 facets

12 vertex figures

P/N, where N=<(s0*s1)^3, s0*s1*s0*s2*(s1*s0)^2*s1*s2*s1*s0, s0*s2*s1*s0*s1*s2*(s1*s0)^2*(s1*s2)^2> of order 8

14 facets

12 vertex figures

P/N, where N=<(s0*s1)^3, s0*s1*s0*s2*s1*s0*s1*s2*s1*s0> of order 12

10 facets

8 vertex figures

Representations

Permutation Representation (GAP) 
s0 := (2,3)(4,5)(6,9)(7,8);;
s1 := ( 1, 2)( 4, 7)( 5, 8)( 9,10);;
s2 := (2,4)(3,5)(6,7)(8,9);;
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, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2, 
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) 
s0 := Sym(10)!(2,3)(4,5)(6,9)(7,8);
s1 := Sym(10)!( 1, 2)( 4, 7)( 5, 8)( 9,10);
s2 := Sym(10)!(2,4)(3,5)(6,7)(8,9);
poly := sub<Sym(10)|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, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2, 
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1 >;

References

None.

to this polytope.

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