Part of the Atlas of Small Regular Polytopes

Polytope of Type {3,2,6,9}

Atlas Canonical Name {3,2,6,9}*648

Overview

Group
SmallGroup(648,554)
Rank
5
Schläfli Type
{3,2,6,9}
Vertices, edges, …
3, 3, 6, 27, 9
Order of s0s1s2s3s4
18
Order of s0s1s2s3s4s3s2s1
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

Representations

Permutation Representation (GAP) 
s0 := (2,3);;
s1 := (1,2);;
s2 := ( 7, 8)(10,11)(13,14)(16,17)(19,20)(22,23)(25,26)(27,28)(29,30);;
s3 := ( 4, 7)( 5,13)( 6,10)( 9,19)(11,14)(12,16)(15,25)(17,20)(18,22)(21,29)(23,26)(24,27)(28,30);;
s4 := ( 4, 5)( 6, 9)( 7,11)( 8,10)(12,15)(13,17)(14,16)(18,21)(19,23)(20,22)(25,28)(26,27)(29,30);;
poly := Group([s0,s1,s2,s3,s4]);;
Finitely Presented Group Representation (GAP) 
F := FreeGroup("s0","s1","s2","s3","s4");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  s4 := F.5;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s4*s4, s0*s2*s0*s2, 
s1*s2*s1*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s0*s4*s0*s4, s1*s4*s1*s4, s2*s4*s2*s4, 
s0*s1*s0*s1*s0*s1, s2*s3*s4*s2*s3*s2*s3*s4*s2*s3, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4 ];;
poly := F / rels;;
Permutation Representation (Magma) 
s0 := Sym(30)!(2,3);
s1 := Sym(30)!(1,2);
s2 := Sym(30)!( 7, 8)(10,11)(13,14)(16,17)(19,20)(22,23)(25,26)(27,28)(29,30);
s3 := Sym(30)!( 4, 7)( 5,13)( 6,10)( 9,19)(11,14)(12,16)(15,25)(17,20)(18,22)(21,29)(23,26)(24,27)(28,30);
s4 := Sym(30)!( 4, 5)( 6, 9)( 7,11)( 8,10)(12,15)(13,17)(14,16)(18,21)(19,23)(20,22)(25,28)(26,27)(29,30);
poly := sub<Sym(30)|s0,s1,s2,s3,s4>;
Finitely Presented Group Representation (Magma) 
poly<s0,s1,s2,s3,s4> := Group< s0,s1,s2,s3,s4 | s0*s0, s1*s1, s2*s2, 
s3*s3, s4*s4, s0*s2*s0*s2, s1*s2*s1*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s0*s4*s0*s4, 
s1*s4*s1*s4, s2*s4*s2*s4, s0*s1*s0*s1*s0*s1, 
s2*s3*s4*s2*s3*s2*s3*s4*s2*s3, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4 >;