Part of the Atlas of Small Regular Polytopes

Polytope of Type {6,3}

Atlas Canonical Name {6,3}*432

▶ Play as a twisty puzzle

Overview

Group
SmallGroup(432,523)
Rank
3
Schläfli Type
{6,3}
Vertices, edges, …
72, 108, 36
Order of s0s1s2
12
Order of s0s1s2s1
6
Also known as
{6,3}(6,0), {6,3}12. if this polytope has another name.

Special Properties

  • Toroidal
  • Locally Spherical
  • Orientable

Quotients maximal quotients in bold

3-fold

4-fold

9-fold

12-fold

18-fold

36-fold

Covers minimal covers in bold

2-fold

3-fold

4-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*s2*(s1*s0)^2)^2*s1*s2*s1*s0*s1> of order 2

18 facets

36 vertex figures

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

20 facets

36 vertex figures

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

14 facets

24 vertex figures

P/N, where N=<((s1*s0)^2*s1*s2)^2> of order 3

12 facets

24 vertex figures

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

9 facets

18 vertex figures

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

9 facets

18 vertex figures

P/N, where N=<((s1*s0)^2*s1*s2)^2, (s0*s1)^2*(s2*(s1*s0)^2)^2*s2*s1> of order 6

6 facets

12 vertex figures

P/N, where N=<(s0*s1)^3, s0*s2*(s1*s0)^2*s1*s2> of order 6

8 facets

12 vertex figures

Representations

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

References

None.

to this polytope.

Twisty Puzzle