Part of the Atlas of Small Regular Polytopes

Polytope of Type {6,6}

Atlas Canonical Name {6,6}*660

▶ Play as a twisty puzzle

Overview

Group
SmallGroup(660,13)
Rank
3
Schläfli Type
{6,6}
Vertices, edges, …
55, 165, 55
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
  • Self-Dual

Quotients maximal quotients in bold

No regular quotients.

Covers minimal covers in bold

2-fold

Irregular Quotients of which this is a minimal cover

None.

Representations

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

References

None.

to this polytope.

Twisty Puzzle