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Polytope of Type {18,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {18,4}*1296b
if this polytope has a name.
Group : SmallGroup(1296,2908)
Rank : 3
Schlafli Type : {18,4}
Number of vertices, edges, etc : 162, 324, 36
Order of s₀s₁s₂ : 36
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 : {6,4}*432b
   9-fold quotients : {18,4}*144a, {6,4}*144
   18-fold quotients : {18,2}*72, {6,4}*72
   27-fold quotients : {6,4}*48a
   36-fold quotients : {9,2}*36
   54-fold quotients : {6,2}*24
   81-fold quotients : {2,4}*16
   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)(10,19)(11,21)(12,20)(13,26)(14,25)(15,27)
(16,23)(17,22)(18,24)(28,55)(29,57)(30,56)(31,62)(32,61)(33,63)(34,59)(35,58)
(36,60)(37,73)(38,75)(39,74)(40,80)(41,79)(42,81)(43,77)(44,76)(45,78)(46,64)
(47,66)(48,65)(49,71)(50,70)(51,72)(52,68)(53,67)(54,69);;
s1 := ( 1,31)( 2,33)( 3,32)( 4,28)( 5,30)( 6,29)( 7,35)( 8,34)( 9,36)(10,40)
(11,42)(12,41)(13,37)(14,39)(15,38)(16,44)(17,43)(18,45)(19,49)(20,51)(21,50)
(22,46)(23,48)(24,47)(25,53)(26,52)(27,54)(55,58)(56,60)(57,59)(61,62)(64,67)
(65,69)(66,68)(70,71)(73,76)(74,78)(75,77)(79,80);;
s2 := (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);;
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, 
s2*s0*s1*s0*s1*s2*s1*s0*s1*s2*s0*s1*s0*s1*s2*s1*s0*s1, 
s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;

Permutation Representation (Magma) :

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

References : None.
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