Part of the Atlas of Small Regular Polytopes

Polytope of Type {6,3,6}

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

Overview

Group
SmallGroup(648,555)
Rank
4
Schläfli Type
{6,3,6}
Vertices, edges, …
18, 27, 27, 6
Order of s0s1s2s3
6
Order of s0s1s2s3s2s1
2
Also known as
6T4(3,0)(1,1). if this polytope has another name.

Special Properties

  • Universal
  • Locally Toroidal
  • Orientable
  • Flat

Quotients maximal quotients in bold

3-fold

9-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

6 facets

  • 6 of 3-fold non-regular quotient of {6,3}*108

6 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 := ( 4, 7)( 5, 8)( 6, 9)(10,20)(11,21)(12,19)(13,26)(14,27)(15,25)(16,23)(17,24)(18,22);;
s2 := ( 1,13)( 2,14)( 3,15)( 4,10)( 5,11)( 6,12)( 7,16)( 8,17)( 9,18)(19,22)(20,23)(21,24);;
s3 := ( 4, 7)( 5, 8)( 6, 9)(13,16)(14,17)(15,18)(22,25)(23,26)(24,27);;
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, s1*s2*s1*s2*s1*s2, 
s3*s1*s2*s3*s2*s3*s1*s2*s3*s2, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s1*s2*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)!( 4, 7)( 5, 8)( 6, 9)(10,20)(11,21)(12,19)(13,26)(14,27)(15,25)(16,23)(17,24)(18,22);
s2 := Sym(27)!( 1,13)( 2,14)( 3,15)( 4,10)( 5,11)( 6,12)( 7,16)( 8,17)( 9,18)(19,22)(20,23)(21,24);
s3 := Sym(27)!( 4, 7)( 5, 8)( 6, 9)(13,16)(14,17)(15,18)(22,25)(23,26)(24,27);
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, 
s1*s2*s1*s2*s1*s2, s3*s1*s2*s3*s2*s3*s1*s2*s3*s2, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s1*s2*s1 >;

References

  1. Theorem 11C7,11C8, McMullen P., Schulte, E.; Abstract Regular Polytopes (Cambridge University Press, 2002)

to this polytope.