Part of the Atlas of Small Regular Polytopes

Polytope of Type {6,9}

Atlas Canonical Name {6,9}*108

▶ Play as a twisty puzzle

Overview

Group
SmallGroup(108,16)
Rank
3
Schläfli Type
{6,9}
Vertices, edges, …
6, 27, 9
Order of s0s1s2
18
Order of s0s1s2s1
6
Also known as
if this polytope has a name.

Special Properties

  • Compact Hyperbolic Quotient
  • Locally Spherical
  • Orientable
  • Flat

Quotients maximal quotients in bold

3-fold

9-fold

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

13-fold

14-fold

15-fold

16-fold

17-fold

18-fold

Irregular Quotients of which this is a minimal cover

None.

Representations

Permutation Representation (GAP) 
s0 := ( 4, 5)( 7, 8)(10,11)(13,14)(16,17)(19,20)(22,23)(24,25)(26,27);;
s1 := ( 1, 4)( 2,10)( 3, 7)( 6,16)( 8,11)( 9,13)(12,22)(14,17)(15,19)(18,26)(20,23)(21,24)(25,27);;
s2 := ( 1, 2)( 3, 6)( 4, 8)( 5, 7)( 9,12)(10,14)(11,13)(15,18)(16,20)(17,19)(22,25)(23,24)(26,27);;
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*s2*s0*s1*s0*s1*s2*s0*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;
Permutation Representation (Magma) 
s0 := Sym(27)!( 4, 5)( 7, 8)(10,11)(13,14)(16,17)(19,20)(22,23)(24,25)(26,27);
s1 := Sym(27)!( 1, 4)( 2,10)( 3, 7)( 6,16)( 8,11)( 9,13)(12,22)(14,17)(15,19)(18,26)(20,23)(21,24)(25,27);
s2 := Sym(27)!( 1, 2)( 3, 6)( 4, 8)( 5, 7)( 9,12)(10,14)(11,13)(15,18)(16,20)(17,19)(22,25)(23,24)(26,27);
poly := sub<Sym(27)|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*s2*s0*s1*s0*s1*s2*s0*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;

References

None.

to this polytope.

Twisty Puzzle