Polytope of Type {36,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {36,4}*1296
if this polytope has a name.
Group : SmallGroup(1296,2908)
Rank : 3
Schlafli Type : {36,4}
Number of vertices, edges, etc : 162, 324, 18
Order of s₀s₁s₂ : 18
Order of s₀s₁s₂s₁ : 6
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
   Skewing Operation
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {12,4}*432b
   9-fold quotients : {4,4}*144
   18-fold quotients : {18,2}*72, {4,4}*72
   36-fold quotients : {9,2}*36
   54-fold quotients : {6,2}*24
   108-fold quotients : {3,2}*12
   162-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
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