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Polytope of Type {3,2,3,6,9}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,2,3,6,9}*1944
if this polytope has a name.
Group : SmallGroup(1944,2340)
Rank : 6
Schlafli Type : {3,2,3,6,9}
Number of vertices, edges, etc : 3, 3, 3, 9, 27, 9
Order of s0s1s2s3s4s5 : 9
Order of s0s1s2s3s4s5s4s3s2s1 : 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 : {3,2,3,2,9}*648, {3,2,3,6,3}*648
   9-fold quotients : {3,2,3,2,3}*216
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :
s0 := (2,3);;
s1 := (1,2);;
s2 := ( 5, 6)( 8, 9)(11,12)(13,22)(14,24)(15,23)(16,25)(17,27)(18,26)(19,28)
(20,30)(21,29)(32,33)(35,36)(38,39)(40,49)(41,51)(42,50)(43,52)(44,54)(45,53)
(46,55)(47,57)(48,56)(59,60)(62,63)(65,66)(67,76)(68,78)(69,77)(70,79)(71,81)
(72,80)(73,82)(74,84)(75,83);;
s3 := ( 4,13)( 5,15)( 6,14)( 7,16)( 8,18)( 9,17)(10,19)(11,21)(12,20)(23,24)
(26,27)(29,30)(31,40)(32,42)(33,41)(34,43)(35,45)(36,44)(37,46)(38,48)(39,47)
(50,51)(53,54)(56,57)(58,67)(59,69)(60,68)(61,70)(62,72)(63,71)(64,73)(65,75)
(66,74)(77,78)(80,81)(83,84);;
s4 := ( 4,31)( 5,33)( 6,32)( 7,37)( 8,39)( 9,38)(10,34)(11,36)(12,35)(13,41)
(14,40)(15,42)(16,47)(17,46)(18,48)(19,44)(20,43)(21,45)(22,51)(23,50)(24,49)
(25,57)(26,56)(27,55)(28,54)(29,53)(30,52)(58,61)(59,63)(60,62)(65,66)(67,71)
(68,70)(69,72)(73,74)(76,81)(77,80)(78,79)(82,84);;
s5 := ( 5, 6)( 7,10)( 8,12)( 9,11)(14,15)(16,19)(17,21)(18,20)(23,24)(25,28)
(26,30)(27,29)(31,61)(32,63)(33,62)(34,58)(35,60)(36,59)(37,64)(38,66)(39,65)
(40,70)(41,72)(42,71)(43,67)(44,69)(45,68)(46,73)(47,75)(48,74)(49,79)(50,81)
(51,80)(52,76)(53,78)(54,77)(55,82)(56,84)(57,83);;
poly := Group([s0,s1,s2,s3,s4,s5]);;
 
Finitely Presented Group Representation (GAP) :
F := FreeGroup("s0","s1","s2","s3","s4","s5");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  s4 := F.5;;  s5 := F.6;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s4*s4, s5*s5, 
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*s5*s0*s5, s1*s5*s1*s5, 
s2*s5*s2*s5, s3*s5*s3*s5, s0*s1*s0*s1*s0*s1, 
s2*s3*s2*s3*s2*s3, s4*s2*s3*s4*s3*s4*s2*s3*s4*s3, 
s5*s3*s4*s3*s4*s5*s3*s4*s3*s4, s4*s5*s4*s5*s4*s5*s4*s5*s4*s5*s4*s5*s4*s5*s4*s5*s4*s5 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(84)!(2,3);
s1 := Sym(84)!(1,2);
s2 := Sym(84)!( 5, 6)( 8, 9)(11,12)(13,22)(14,24)(15,23)(16,25)(17,27)(18,26)
(19,28)(20,30)(21,29)(32,33)(35,36)(38,39)(40,49)(41,51)(42,50)(43,52)(44,54)
(45,53)(46,55)(47,57)(48,56)(59,60)(62,63)(65,66)(67,76)(68,78)(69,77)(70,79)
(71,81)(72,80)(73,82)(74,84)(75,83);
s3 := Sym(84)!( 4,13)( 5,15)( 6,14)( 7,16)( 8,18)( 9,17)(10,19)(11,21)(12,20)
(23,24)(26,27)(29,30)(31,40)(32,42)(33,41)(34,43)(35,45)(36,44)(37,46)(38,48)
(39,47)(50,51)(53,54)(56,57)(58,67)(59,69)(60,68)(61,70)(62,72)(63,71)(64,73)
(65,75)(66,74)(77,78)(80,81)(83,84);
s4 := Sym(84)!( 4,31)( 5,33)( 6,32)( 7,37)( 8,39)( 9,38)(10,34)(11,36)(12,35)
(13,41)(14,40)(15,42)(16,47)(17,46)(18,48)(19,44)(20,43)(21,45)(22,51)(23,50)
(24,49)(25,57)(26,56)(27,55)(28,54)(29,53)(30,52)(58,61)(59,63)(60,62)(65,66)
(67,71)(68,70)(69,72)(73,74)(76,81)(77,80)(78,79)(82,84);
s5 := Sym(84)!( 5, 6)( 7,10)( 8,12)( 9,11)(14,15)(16,19)(17,21)(18,20)(23,24)
(25,28)(26,30)(27,29)(31,61)(32,63)(33,62)(34,58)(35,60)(36,59)(37,64)(38,66)
(39,65)(40,70)(41,72)(42,71)(43,67)(44,69)(45,68)(46,73)(47,75)(48,74)(49,79)
(50,81)(51,80)(52,76)(53,78)(54,77)(55,82)(56,84)(57,83);
poly := sub<Sym(84)|s0,s1,s2,s3,s4,s5>;
 
Finitely Presented Group Representation (Magma) :
poly<s0,s1,s2,s3,s4,s5> := Group< s0,s1,s2,s3,s4,s5 | s0*s0, s1*s1, s2*s2, 
s3*s3, s4*s4, s5*s5, 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*s5*s0*s5, 
s1*s5*s1*s5, s2*s5*s2*s5, s3*s5*s3*s5, 
s0*s1*s0*s1*s0*s1, s2*s3*s2*s3*s2*s3, 
s4*s2*s3*s4*s3*s4*s2*s3*s4*s3, s5*s3*s4*s3*s4*s5*s3*s4*s3*s4, 
s4*s5*s4*s5*s4*s5*s4*s5*s4*s5*s4*s5*s4*s5*s4*s5*s4*s5 >; 
 

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