Polytope of Type {4,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,4}*648
Also Known As : {4,4}_(9,0), {4,4|9}. if this polytope has another name.
Group : SmallGroup(648,252)
Rank : 3
Schlafli Type : {4,4}
Number of vertices, edges, etc : 81, 162, 81
Order of s₀s₁s₂ : 18
Order of s₀s₁s₂s₁ : 9
Special Properties :
   Toroidal
   Locally Spherical
   Orientable
   Self-Dual
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
   Halving Operation
   Skewing Operation
Facet Of :
   {4,4,2} of size 1296
Vertex Figure Of :
   {2,4,4} of size 1296
Quotients (Maximal Quotients in Boldface) :
   9-fold quotients : {4,4}*72
Covers (Minimal Covers in Boldface) :
   2-fold covers : {4,4}*1296
   3-fold covers : {4,12}*1944a, {12,4}*1944a
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