Part of the Atlas of Small Regular Polytopes

Polytope of Type {6,3,6}

Atlas Canonical Name {6,3,6}*864b

Overview

Group
SmallGroup(864,4673)
Rank
4
Schläfli Type
{6,3,6}
Vertices, edges, …
24, 36, 36, 6
Order of s0s1s2s3
12
Order of s0s1s2s3s2s1
2
Also known as
6T4(2,2)(1,1). if this polytope has another name.

Special Properties

  • Universal
  • Locally Toroidal
  • Orientable
  • Flat

Quotients maximal quotients in bold

3-fold

4-fold

9-fold

12-fold

18-fold

36-fold

Covers minimal covers in bold

2-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*s0*s2*s1*s0*s1*s2> of order 2

6 facets

  • 6 of 2-fold non-regular quotient of {6,3}*144

12 vertex figures

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

6 facets

  • 6 of 3-fold non-regular quotient of {6,3}*144

8 vertex figures

P/N, where N=<s0*s1*s2*(s1*s0)^2*s2*s1> of order 4

6 facets

  • 6 of 4-fold non-regular quotient of {6,3}*144

6 vertex figures

Representations

Permutation Representation (GAP) 
s0 := ( 2, 3)( 6, 7)(10,11)(14,15)(18,19)(22,23)(26,27)(30,31)(34,35);;
s1 := ( 3, 4)( 5, 9)( 6,10)( 7,12)( 8,11)(13,25)(14,26)(15,28)(16,27)(17,33)(18,34)(19,36)(20,35)(21,29)(22,30)(23,32)(24,31);;
s2 := ( 1,20)( 2,18)( 3,19)( 4,17)( 5,16)( 6,14)( 7,15)( 8,13)( 9,24)(10,22)(11,23)(12,21)(25,32)(26,30)(27,31)(28,29)(33,36);;
s3 := ( 5, 9)( 6,10)( 7,11)( 8,12)(17,21)(18,22)(19,23)(20,24)(29,33)(30,34)(31,35)(32,36);;
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*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) 
s0 := Sym(36)!( 2, 3)( 6, 7)(10,11)(14,15)(18,19)(22,23)(26,27)(30,31)(34,35);
s1 := Sym(36)!( 3, 4)( 5, 9)( 6,10)( 7,12)( 8,11)(13,25)(14,26)(15,28)(16,27)(17,33)(18,34)(19,36)(20,35)(21,29)(22,30)(23,32)(24,31);
s2 := Sym(36)!( 1,20)( 2,18)( 3,19)( 4,17)( 5,16)( 6,14)( 7,15)( 8,13)( 9,24)(10,22)(11,23)(12,21)(25,32)(26,30)(27,31)(28,29)(33,36);
s3 := Sym(36)!( 5, 9)( 6,10)( 7,11)( 8,12)(17,21)(18,22)(19,23)(20,24)(29,33)(30,34)(31,35)(32,36);
poly := sub<Sym(36)|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*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1 >;

References

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

to this polytope.