Part of the Atlas of Small Regular Polytopes

Polytope of Type {3,6}

Atlas Canonical Name {3,6}*432

▶ Play as a twisty puzzle

Overview

Group
SmallGroup(432,523)
Rank
3
Schläfli Type
{3,6}
Vertices, edges, …
36, 108, 72
Order of s0s1s2
12
Order of s0s1s2s1
6
Also known as
{3,6}(6,0), {3,6}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*s0*s2)^5*s1*s2> of order 2

36 facets

18 vertex figures

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

36 facets

20 vertex figures

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

24 facets

14 vertex figures

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

24 facets

12 vertex figures

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

18 facets

9 vertex figures

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

18 facets

9 vertex figures

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

12 facets

6 vertex figures

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

12 facets

8 vertex figures

Representations

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

References

None.

to this polytope.

Twisty Puzzle