Part of the Atlas of Small Regular Polytopes

Polytope of Type {4,4}

Atlas Canonical Name {4,4}*648

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Overview

Group
SmallGroup(648,252)
Rank
3
Schläfli Type
{4,4}
Vertices, edges, …
81, 162, 81
Order of s0s1s2
18
Order of s0s1s2s1
9
Also known as
{4,4}(9,0), {4,4|9}. if this polytope has another name.

Special Properties

  • Toroidal
  • Locally Spherical
  • Orientable
  • Self-Dual

Quotients maximal quotients in bold

9-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)^3> of order 3

27 facets

27 vertex figures

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

27 facets

27 vertex figures

P/N, where N=<(s0*s1)^2*s2*s1*s0*s2*s1*s2> of order 9

9 facets

9 vertex figures

P/N, where N=<(s0*s1*s2*s1)^3, s0*(s2*s1*s0*s1)^2*s2*s1*s0*s2*s1> of order 9

9 facets

9 vertex figures

Representations

Permutation Representation (GAP) 
s0 := ( 2, 5)( 3, 9)( 4, 7)(10,13)(11,17)(15,18)(19,25)(21,24)(23,26)(28,61)(29,56)(30,60)(31,58)(32,62)(33,57)(34,55)(35,59)(36,63)(37,64)(38,68)(39,72)(40,70)(41,65)(42,69)(43,67)(44,71)(45,66)(46,76)(47,80)(48,75)(49,73)(50,77)(51,81)(52,79)(53,74)(54,78);;
s1 := ( 2, 3)( 4, 6)( 7, 8)(10,30)(11,29)(12,28)(13,32)(14,31)(15,33)(16,34)(17,36)(18,35)(19,56)(20,55)(21,57)(22,58)(23,60)(24,59)(25,63)(26,62)(27,61)(37,38)(41,42)(43,45)(46,64)(47,66)(48,65)(49,69)(50,68)(51,67)(52,71)(53,70)(54,72)(73,75)(76,77)(80,81);;
s2 := ( 1,22)( 2,21)( 3,26)( 4,25)( 5,24)( 6,20)( 7,19)( 8,27)( 9,23)(11,18)(12,14)(15,17)(28,49)(29,48)(30,53)(31,52)(32,51)(33,47)(34,46)(35,54)(36,50)(38,45)(39,41)(42,44)(55,76)(56,75)(57,80)(58,79)(59,78)(60,74)(61,73)(62,81)(63,77)(65,72)(66,68)(69,71);;
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, s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) 
s0 := Sym(81)!( 2, 5)( 3, 9)( 4, 7)(10,13)(11,17)(15,18)(19,25)(21,24)(23,26)(28,61)(29,56)(30,60)(31,58)(32,62)(33,57)(34,55)(35,59)(36,63)(37,64)(38,68)(39,72)(40,70)(41,65)(42,69)(43,67)(44,71)(45,66)(46,76)(47,80)(48,75)(49,73)(50,77)(51,81)(52,79)(53,74)(54,78);
s1 := Sym(81)!( 2, 3)( 4, 6)( 7, 8)(10,30)(11,29)(12,28)(13,32)(14,31)(15,33)(16,34)(17,36)(18,35)(19,56)(20,55)(21,57)(22,58)(23,60)(24,59)(25,63)(26,62)(27,61)(37,38)(41,42)(43,45)(46,64)(47,66)(48,65)(49,69)(50,68)(51,67)(52,71)(53,70)(54,72)(73,75)(76,77)(80,81);
s2 := Sym(81)!( 1,22)( 2,21)( 3,26)( 4,25)( 5,24)( 6,20)( 7,19)( 8,27)( 9,23)(11,18)(12,14)(15,17)(28,49)(29,48)(30,53)(31,52)(32,51)(33,47)(34,46)(35,54)(36,50)(38,45)(39,41)(42,44)(55,76)(56,75)(57,80)(58,79)(59,78)(60,74)(61,73)(62,81)(63,77)(65,72)(66,68)(69,71);
poly := sub<Sym(81)|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, s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1 >;

References

None.

to this polytope.

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