Part of the Atlas of Small Regular Polytopes

Polytope of Type {2,6,9}

Atlas Canonical Name {2,6,9}*216

Overview

Group
SmallGroup(216,101)
Rank
4
Schläfli Type
{2,6,9}
Vertices, edges, …
2, 6, 27, 9
Order of s0s1s2s3
18
Order of s0s1s2s3s2s1
2
Also known as
if this polytope has a name.

Special Properties

  • Degenerate
  • Universal
  • 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

Representations

Permutation Representation (GAP) 
s0 := (1,2);;
s1 := ( 6, 7)( 9,10)(12,13)(15,16)(18,19)(21,22)(24,25)(26,27)(28,29);;
s2 := ( 3, 6)( 4,12)( 5, 9)( 8,18)(10,13)(11,15)(14,24)(16,19)(17,21)(20,28)(22,25)(23,26)(27,29);;
s3 := ( 3, 4)( 5, 8)( 6,10)( 7, 9)(11,14)(12,16)(13,15)(17,20)(18,22)(19,21)(24,27)(25,26)(28,29);;
poly := Group([s0,s1,s2,s3]);;
Finitely Presented Group Representation (GAP) 
F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s1*s0*s1, 
s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s1*s2*s3*s1*s2*s1*s2*s3*s1*s2, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;
Permutation Representation (Magma) 
s0 := Sym(29)!(1,2);
s1 := Sym(29)!( 6, 7)( 9,10)(12,13)(15,16)(18,19)(21,22)(24,25)(26,27)(28,29);
s2 := Sym(29)!( 3, 6)( 4,12)( 5, 9)( 8,18)(10,13)(11,15)(14,24)(16,19)(17,21)(20,28)(22,25)(23,26)(27,29);
s3 := Sym(29)!( 3, 4)( 5, 8)( 6,10)( 7, 9)(11,14)(12,16)(13,15)(17,20)(18,22)(19,21)(24,27)(25,26)(28,29);
poly := sub<Sym(29)|s0,s1,s2,s3>;
Finitely Presented Group Representation (Magma) 
poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2, 
s3*s3, s0*s1*s0*s1, s0*s2*s0*s2, s0*s3*s0*s3, 
s1*s3*s1*s3, s1*s2*s3*s1*s2*s1*s2*s3*s1*s2, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >;