Part of the Atlas of Small Regular Polytopes

Polytope of Type {4,4}

Atlas Canonical Name {4,4}*576

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Overview

Group
SmallGroup(576,5296)
Rank
3
Schläfli Type
{4,4}
Vertices, edges, …
72, 144, 72
Order of s0s1s2
12
Order of s0s1s2s1
12
Also known as
{4,4}(6,6), {4,4}12. if this polytope has another name.

Special Properties

  • Toroidal
  • Locally Spherical
  • Orientable
  • Self-Dual

Quotients maximal quotients in bold

2-fold

4-fold

8-fold

9-fold

18-fold

36-fold

72-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*s2*s1)^5*s0*s1*s2> of order 2

36 facets

36 vertex figures

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

38 facets

36 vertex figures

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

36 facets

36 vertex figures

P/N, where N=<s0*s1*(s2*s1*s0)^5*s2> of order 2

36 facets

38 vertex figures

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

24 facets

24 vertex figures

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

24 facets

24 vertex figures

P/N, where N=<s1*(s2*s1*s0)^4*s2*s1*s2, s0*s1*(s2*s1*s0)^5*s2*s1> of order 4

18 facets

19 vertex figures

P/N, where N=<(s0*s2*s1)^6, (s0*s1)^2*(s2*s1*s0)^4*s2*s1*s2> of order 4

18 facets

18 vertex figures

P/N, where N=<(s0*s1)^2, s0*s1*(s2*s1*s0)^5*s2> of order 4

19 facets

19 vertex figures

P/N, where N=<(s0*s1)^2, (s0*s1*s2*s1)^5*s0*s1*s2> of order 4

19 facets

18 vertex figures

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

18 facets

18 vertex figures

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

12 facets

14 vertex figures

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

14 facets

12 vertex figures

P/N, where N=<s0*(s1*s0*s2)^3*s1*s2, s0*s1*(s2*s1*s0)^5*s2> of order 6

12 facets

14 vertex figures

P/N, where N=<s0*(s1*s0*s2)^3*s1*s2, s0*s2*s1*s0*s1*(s2*s1*s0)^4*s2*s1> of order 6

12 facets

12 vertex figures

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

14 facets

12 vertex figures

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

12 facets

12 vertex figures

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

12 facets

12 vertex figures

P/N, where N=<(s0*s1)^2, s1*(s2*s1*s0)^2*s2*s1*s2, s0*s1*(s2*s1*s0)^5*s2> of order 12

7 facets

7 vertex figures

P/N, where N=<(s0*s1*s2*s1)^2*s0*s2*s1, s0*(s2*s1*s0*s1)^2*s2*s1> of order 12

6 facets

6 vertex figures

P/N, where N=<(s0*s1)^2, (s1*s2*s1*s0)^2*(s1*s2)^2> of order 12

7 facets

7 vertex figures

Representations

Permutation Representation (GAP) 
s0 := ( 4, 7)( 5, 8)( 6, 9)(13,16)(14,17)(15,18)(19,28)(20,29)(21,30)(22,34)(23,35)(24,36)(25,31)(26,32)(27,33)(40,43)(41,44)(42,45)(49,52)(50,53)(51,54)(55,64)(56,65)(57,66)(58,70)(59,71)(60,72)(61,67)(62,68)(63,69);;
s1 := ( 2, 4)( 3, 7)( 6, 8)(11,13)(12,16)(15,17)(20,22)(21,25)(24,26)(29,31)(30,34)(33,35)(37,55)(38,58)(39,61)(40,56)(41,59)(42,62)(43,57)(44,60)(45,63)(46,64)(47,67)(48,70)(49,65)(50,68)(51,71)(52,66)(53,69)(54,72);;
s2 := ( 1,47)( 2,46)( 3,48)( 4,50)( 5,49)( 6,51)( 7,53)( 8,52)( 9,54)(10,38)(11,37)(12,39)(13,41)(14,40)(15,42)(16,44)(17,43)(18,45)(19,56)(20,55)(21,57)(22,59)(23,58)(24,60)(25,62)(26,61)(27,63)(28,65)(29,64)(30,66)(31,68)(32,67)(33,69)(34,71)(35,70)(36,72);;
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*s0*s1, 
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(72)!( 4, 7)( 5, 8)( 6, 9)(13,16)(14,17)(15,18)(19,28)(20,29)(21,30)(22,34)(23,35)(24,36)(25,31)(26,32)(27,33)(40,43)(41,44)(42,45)(49,52)(50,53)(51,54)(55,64)(56,65)(57,66)(58,70)(59,71)(60,72)(61,67)(62,68)(63,69);
s1 := Sym(72)!( 2, 4)( 3, 7)( 6, 8)(11,13)(12,16)(15,17)(20,22)(21,25)(24,26)(29,31)(30,34)(33,35)(37,55)(38,58)(39,61)(40,56)(41,59)(42,62)(43,57)(44,60)(45,63)(46,64)(47,67)(48,70)(49,65)(50,68)(51,71)(52,66)(53,69)(54,72);
s2 := Sym(72)!( 1,47)( 2,46)( 3,48)( 4,50)( 5,49)( 6,51)( 7,53)( 8,52)( 9,54)(10,38)(11,37)(12,39)(13,41)(14,40)(15,42)(16,44)(17,43)(18,45)(19,56)(20,55)(21,57)(22,59)(23,58)(24,60)(25,62)(26,61)(27,63)(28,65)(29,64)(30,66)(31,68)(32,67)(33,69)(34,71)(35,70)(36,72);
poly := sub<Sym(72)|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*s0*s1, 
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.

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