Part of the Atlas of Small Regular Polytopes

Polytope of Type {4,4}

Atlas Canonical Name {4,4}*144

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Overview

Group
SmallGroup(144,186)
Rank
3
Schläfli Type
{4,4}
Vertices, edges, …
18, 36, 18
Order of s0s1s2
6
Order of s0s1s2s1
6
Also known as
{4,4}(3,3), {4,4}6. if this polytope has another name.

Special Properties

  • Toroidal
  • Locally Spherical
  • Orientable
  • Self-Dual

Quotients maximal quotients in bold

2-fold

18-fold

Covers minimal covers in bold

2-fold

3-fold

4-fold

5-fold

6-fold

7-fold

8-fold

9-fold

10-fold

11-fold

12-fold

13-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)^2*s0*s1*s2> of order 2

9 facets

9 vertex figures

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

10 facets

10 vertex figures

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

6 facets

6 vertex figures

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

5 facets

5 vertex figures

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

4 facets

4 vertex figures

Representations

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

References

None.

to this polytope.

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