Part of the Atlas of Small Regular Polytopes

Polytope of Type {3,6}

Atlas Canonical Name {3,6}*576

▶ Play as a twisty puzzle

Overview

Group
SmallGroup(576,5053)
Rank
3
Schläfli Type
{3,6}
Vertices, edges, …
48, 144, 96
Order of s0s1s2
24
Order of s0s1s2s1
6
Also known as
{3,6}(4,4). if this polytope has another name.

Special Properties

  • Toroidal
  • Locally Spherical
  • Orientable

Quotients maximal quotients in bold

3-fold

4-fold

12-fold

16-fold

24-fold

48-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=<s1*(s0*(s2*s1)^2)^2*s0*(s2*s1)^2*s0*s2*s1*s2> of order 2

48 facets

24 vertex figures

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

48 facets

26 vertex figures

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

32 facets

18 vertex figures

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

24 facets

12 vertex figures

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

24 facets

12 vertex figures

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

24 facets

12 vertex figures

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

24 facets

12 vertex figures

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

24 facets

14 vertex figures

P/N, where N=<s0*s1*s0*(s2*s1)^2*s0*s2*s1, s0*s2*s1*s0*(s2*s1)^2*s0*s2*s1*s2, s0*s1*(s2*s1*s0)^4*(s2*s1)^2*s2> of order 8

12 facets

7 vertex figures

P/N, where N=<s1*s0*(s2*s1)^2*s0*s2*s1*s2, s0*s1*(s2*s1*s0)^4*(s2*s1)^2*s2> of order 8

12 facets

6 vertex figures

P/N, where N=<s0*s1*s2*s1*s0*(s2*s1)^2> of order 8

12 facets

6 vertex figures

P/N, where N=<s0*s1*s0*(s2*s1)^2*s0*s2*s1*s2, s0*s1*(s2*s1*s0)^4*(s2*s1)^2> of order 8

12 facets

6 vertex figures

Representations

Permutation Representation (GAP) 
s0 := ( 3, 4)( 5,12)( 6,11)( 7, 9)( 8,10)(13,14)(17,33)(18,34)(19,36)(20,35)(21,44)(22,43)(23,41)(24,42)(25,39)(26,40)(27,38)(28,37)(29,46)(30,45)(31,47)(32,48);;
s1 := ( 1,17)( 2,19)( 3,18)( 4,20)( 5,24)( 6,22)( 7,23)( 8,21)( 9,32)(10,30)(11,31)(12,29)(13,28)(14,26)(15,27)(16,25)(34,35)(37,40)(41,48)(42,46)(43,47)(44,45);;
s2 := ( 1,15)( 2,16)( 3,14)( 4,13)( 5, 6)(11,12)(17,31)(18,32)(19,30)(20,29)(21,22)(27,28)(33,47)(34,48)(35,46)(36,45)(37,38)(43,44);;
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*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) 
s0 := Sym(48)!( 3, 4)( 5,12)( 6,11)( 7, 9)( 8,10)(13,14)(17,33)(18,34)(19,36)(20,35)(21,44)(22,43)(23,41)(24,42)(25,39)(26,40)(27,38)(28,37)(29,46)(30,45)(31,47)(32,48);
s1 := Sym(48)!( 1,17)( 2,19)( 3,18)( 4,20)( 5,24)( 6,22)( 7,23)( 8,21)( 9,32)(10,30)(11,31)(12,29)(13,28)(14,26)(15,27)(16,25)(34,35)(37,40)(41,48)(42,46)(43,47)(44,45);
s2 := Sym(48)!( 1,15)( 2,16)( 3,14)( 4,13)( 5, 6)(11,12)(17,31)(18,32)(19,30)(20,29)(21,22)(27,28)(33,47)(34,48)(35,46)(36,45)(37,38)(43,44);
poly := sub<Sym(48)|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*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1 >;

References

None.

to this polytope.

Twisty Puzzle