Part of the Atlas of Small Regular Polytopes

Polytope of Type {6,3}

Atlas Canonical Name {6,3}*576

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Overview

Group
SmallGroup(576,5053)
Rank
3
Schläfli Type
{6,3}
Vertices, edges, …
96, 144, 48
Order of s0s1s2
24
Order of s0s1s2s1
6
Also known as
{6,3}(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=<s0*s1*s0*(s2*(s1*s0)^2)^2*s2*(s1*s0)^2*s2*s1> of order 2

24 facets

48 vertex figures

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

26 facets

48 vertex figures

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

18 facets

32 vertex figures

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

12 facets

24 vertex figures

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

12 facets

24 vertex figures

P/N, where N=<((s1*s0)^2*s1*s2)^3> of order 4

12 facets

24 vertex figures

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

12 facets

24 vertex figures

P/N, where N=<(s0*s1)^3, s0*s1*s0*(s2*(s1*s0)^2)^2*s2*(s1*s0)^2*s2*s1> of order 4

14 facets

24 vertex figures

P/N, where N=<(s0*s1)^3, s0*s1*(s2*(s1*s0)^2)^2*s2*s1*s0*s1*s2> of order 8

7 facets

12 vertex figures

P/N, where N=<s0*s1*s0*s2*(s1*s0)^2*s2*s1, (s0*s1)^3*(s2*(s1*s0)^2*s1)^2*s2> of order 8

6 facets

12 vertex figures

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

6 facets

12 vertex figures

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

6 facets

12 vertex figures

Representations

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

References

None.

to this polytope.

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