Part of the Atlas of Small Regular Polytopes

Polytope of Type {6,6,3}

Atlas Canonical Name {6,6,3}*648d

Overview

Group
SmallGroup(648,555)
Rank
4
Schläfli Type
{6,6,3}
Vertices, edges, …
18, 54, 27, 3
Order of s0s1s2s3
6
Order of s0s1s2s3s2s1
6
Also known as
if this polytope has a name.

Special Properties

  • Universal
  • Orientable
  • Flat

Quotients maximal quotients in bold

3-fold

9-fold

18-fold

27-fold

Covers minimal covers in bold

2-fold

3-fold

Irregular Quotients of which this is a minimal cover

Click an entry to reveal its facets and vertex figures.

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

3 facets

6 vertex figures

P/N, where N=<s1*s0*s1*s2*s1*s0*s2*s1> of order 3

3 facets

10 vertex figures

Representations

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

References

None.

to this polytope.