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Polytope of Type {3,2,18,9}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,2,18,9}*1944
if this polytope has a name.
Group : SmallGroup(1944,2339)
Rank : 5
Schlafli Type : {3,2,18,9}
Number of vertices, edges, etc : 3, 3, 18, 81, 9
Order of s0s1s2s3s4 : 18
Order of s0s1s2s3s4s3s2s1 : 2
Special Properties :
   Degenerate
   Universal
   Orientable
   Flat
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 : {3,2,6,9}*648
   9-fold quotients : {3,2,2,9}*216, {3,2,6,3}*216
   27-fold quotients : {3,2,2,3}*72
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :
s0 := (2,3);;
s1 := (1,2);;
s2 := ( 7,10)( 8,11)( 9,12)(16,19)(17,20)(18,21)(25,28)(26,29)(27,30)(31,61)
(32,62)(33,63)(34,58)(35,59)(36,60)(37,64)(38,65)(39,66)(40,70)(41,71)(42,72)
(43,67)(44,68)(45,69)(46,73)(47,74)(48,75)(49,79)(50,80)(51,81)(52,76)(53,77)
(54,78)(55,82)(56,83)(57,84);;
s3 := ( 4,31)( 5,33)( 6,32)( 7,37)( 8,39)( 9,38)(10,34)(11,36)(12,35)(13,50)
(14,49)(15,51)(16,56)(17,55)(18,57)(19,53)(20,52)(21,54)(22,41)(23,40)(24,42)
(25,47)(26,46)(27,48)(28,44)(29,43)(30,45)(58,61)(59,63)(60,62)(65,66)(67,80)
(68,79)(69,81)(70,77)(71,76)(72,78)(73,83)(74,82)(75,84);;
s4 := ( 4,13)( 5,15)( 6,14)( 7,19)( 8,21)( 9,20)(10,16)(11,18)(12,17)(22,23)
(25,29)(26,28)(27,30)(31,70)(32,72)(33,71)(34,67)(35,69)(36,68)(37,73)(38,75)
(39,74)(40,61)(41,63)(42,62)(43,58)(44,60)(45,59)(46,64)(47,66)(48,65)(49,80)
(50,79)(51,81)(52,77)(53,76)(54,78)(55,83)(56,82)(57,84);;
poly := Group([s0,s1,s2,s3,s4]);;
 
Finitely Presented Group Representation (GAP) :
F := FreeGroup("s0","s1","s2","s3","s4");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  s4 := F.5;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s4*s4, s0*s2*s0*s2, 
s1*s2*s1*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s0*s4*s0*s4, s1*s4*s1*s4, s2*s4*s2*s4, 
s0*s1*s0*s1*s0*s1, s4*s2*s3*s2*s3*s4*s2*s3*s2*s3, 
s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(84)!(2,3);
s1 := Sym(84)!(1,2);
s2 := Sym(84)!( 7,10)( 8,11)( 9,12)(16,19)(17,20)(18,21)(25,28)(26,29)(27,30)
(31,61)(32,62)(33,63)(34,58)(35,59)(36,60)(37,64)(38,65)(39,66)(40,70)(41,71)
(42,72)(43,67)(44,68)(45,69)(46,73)(47,74)(48,75)(49,79)(50,80)(51,81)(52,76)
(53,77)(54,78)(55,82)(56,83)(57,84);
s3 := Sym(84)!( 4,31)( 5,33)( 6,32)( 7,37)( 8,39)( 9,38)(10,34)(11,36)(12,35)
(13,50)(14,49)(15,51)(16,56)(17,55)(18,57)(19,53)(20,52)(21,54)(22,41)(23,40)
(24,42)(25,47)(26,46)(27,48)(28,44)(29,43)(30,45)(58,61)(59,63)(60,62)(65,66)
(67,80)(68,79)(69,81)(70,77)(71,76)(72,78)(73,83)(74,82)(75,84);
s4 := Sym(84)!( 4,13)( 5,15)( 6,14)( 7,19)( 8,21)( 9,20)(10,16)(11,18)(12,17)
(22,23)(25,29)(26,28)(27,30)(31,70)(32,72)(33,71)(34,67)(35,69)(36,68)(37,73)
(38,75)(39,74)(40,61)(41,63)(42,62)(43,58)(44,60)(45,59)(46,64)(47,66)(48,65)
(49,80)(50,79)(51,81)(52,77)(53,76)(54,78)(55,83)(56,82)(57,84);
poly := sub<Sym(84)|s0,s1,s2,s3,s4>;
 
Finitely Presented Group Representation (Magma) :
poly<s0,s1,s2,s3,s4> := Group< s0,s1,s2,s3,s4 | s0*s0, s1*s1, s2*s2, 
s3*s3, s4*s4, s0*s2*s0*s2, s1*s2*s1*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s0*s4*s0*s4, 
s1*s4*s1*s4, s2*s4*s2*s4, s0*s1*s0*s1*s0*s1, 
s4*s2*s3*s2*s3*s4*s2*s3*s2*s3, s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4 >; 
 

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