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Polytope of Type {4,36}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,36}*1296
if this polytope has a name.
Group : SmallGroup(1296,2908)
Rank : 3
Schlafli Type : {4,36}
Number of vertices, edges, etc : 18, 324, 162
Order of s0s1s2 : 18
Order of s0s1s2s1 : 6
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {4,12}*432b
   9-fold quotients : {4,4}*144
   18-fold quotients : {2,18}*72, {4,4}*72
   36-fold quotients : {2,9}*36
   54-fold quotients : {2,6}*24
   108-fold quotients : {2,3}*12
   162-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :
s0 := (10,28)(11,29)(12,30)(13,31)(14,32)(15,33)(16,34)(17,35)(18,36)(19,55)
(20,56)(21,57)(22,58)(23,59)(24,60)(25,61)(26,62)(27,63)(46,64)(47,65)(48,66)
(49,67)(50,68)(51,69)(52,70)(53,71)(54,72);;
s1 := ( 2, 3)( 4, 8)( 5, 7)( 6, 9)(11,12)(13,17)(14,16)(15,18)(20,21)(22,26)
(23,25)(24,27)(28,55)(29,57)(30,56)(31,62)(32,61)(33,63)(34,59)(35,58)(36,60)
(37,64)(38,66)(39,65)(40,71)(41,70)(42,72)(43,68)(44,67)(45,69)(46,73)(47,75)
(48,74)(49,80)(50,79)(51,81)(52,77)(53,76)(54,78);;
s2 := ( 1,40)( 2,42)( 3,41)( 4,37)( 5,39)( 6,38)( 7,44)( 8,43)( 9,45)(10,13)
(11,15)(12,14)(16,17)(19,67)(20,69)(21,68)(22,64)(23,66)(24,65)(25,71)(26,70)
(27,72)(28,31)(29,33)(30,32)(34,35)(46,58)(47,60)(48,59)(49,55)(50,57)(51,56)
(52,62)(53,61)(54,63)(73,76)(74,78)(75,77)(79,80);;
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, 
s0*s1*s2*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*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, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s0*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(81)!(10,28)(11,29)(12,30)(13,31)(14,32)(15,33)(16,34)(17,35)(18,36)
(19,55)(20,56)(21,57)(22,58)(23,59)(24,60)(25,61)(26,62)(27,63)(46,64)(47,65)
(48,66)(49,67)(50,68)(51,69)(52,70)(53,71)(54,72);
s1 := Sym(81)!( 2, 3)( 4, 8)( 5, 7)( 6, 9)(11,12)(13,17)(14,16)(15,18)(20,21)
(22,26)(23,25)(24,27)(28,55)(29,57)(30,56)(31,62)(32,61)(33,63)(34,59)(35,58)
(36,60)(37,64)(38,66)(39,65)(40,71)(41,70)(42,72)(43,68)(44,67)(45,69)(46,73)
(47,75)(48,74)(49,80)(50,79)(51,81)(52,77)(53,76)(54,78);
s2 := Sym(81)!( 1,40)( 2,42)( 3,41)( 4,37)( 5,39)( 6,38)( 7,44)( 8,43)( 9,45)
(10,13)(11,15)(12,14)(16,17)(19,67)(20,69)(21,68)(22,64)(23,66)(24,65)(25,71)
(26,70)(27,72)(28,31)(29,33)(30,32)(34,35)(46,58)(47,60)(48,59)(49,55)(50,57)
(51,56)(52,62)(53,61)(54,63)(73,76)(74,78)(75,77)(79,80);
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, 
s0*s1*s2*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*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, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s0*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2 >; 
 
References : None.
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