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

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,2,3,6,9}*1296
if this polytope has a name.
Group : SmallGroup(1296,1858)
Rank : 6
Schlafli Type : {2,2,3,6,9}
Number of vertices, edges, etc : 2, 2, 3, 9, 27, 9
Order of s0s1s2s3s4s5 : 18
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 : {2,2,3,2,9}*432, {2,2,3,6,3}*432
   9-fold quotients : {2,2,3,2,3}*144
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :
s0 := (1,2);;
s1 := (3,4);;
s2 := ( 6, 7)( 9,10)(12,13)(14,23)(15,25)(16,24)(17,26)(18,28)(19,27)(20,29)
(21,31)(22,30)(33,34)(36,37)(39,40)(41,50)(42,52)(43,51)(44,53)(45,55)(46,54)
(47,56)(48,58)(49,57)(60,61)(63,64)(66,67)(68,77)(69,79)(70,78)(71,80)(72,82)
(73,81)(74,83)(75,85)(76,84);;
s3 := ( 5,14)( 6,16)( 7,15)( 8,17)( 9,19)(10,18)(11,20)(12,22)(13,21)(24,25)
(27,28)(30,31)(32,41)(33,43)(34,42)(35,44)(36,46)(37,45)(38,47)(39,49)(40,48)
(51,52)(54,55)(57,58)(59,68)(60,70)(61,69)(62,71)(63,73)(64,72)(65,74)(66,76)
(67,75)(78,79)(81,82)(84,85);;
s4 := ( 5,32)( 6,34)( 7,33)( 8,38)( 9,40)(10,39)(11,35)(12,37)(13,36)(14,42)
(15,41)(16,43)(17,48)(18,47)(19,49)(20,45)(21,44)(22,46)(23,52)(24,51)(25,50)
(26,58)(27,57)(28,56)(29,55)(30,54)(31,53)(59,62)(60,64)(61,63)(66,67)(68,72)
(69,71)(70,73)(74,75)(77,82)(78,81)(79,80)(83,85);;
s5 := ( 6, 7)( 8,11)( 9,13)(10,12)(15,16)(17,20)(18,22)(19,21)(24,25)(26,29)
(27,31)(28,30)(32,62)(33,64)(34,63)(35,59)(36,61)(37,60)(38,65)(39,67)(40,66)
(41,71)(42,73)(43,72)(44,68)(45,70)(46,69)(47,74)(48,76)(49,75)(50,80)(51,82)
(52,81)(53,77)(54,79)(55,78)(56,83)(57,85)(58,84);;
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*s1*s0*s1, 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, 
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(85)!(1,2);
s1 := Sym(85)!(3,4);
s2 := Sym(85)!( 6, 7)( 9,10)(12,13)(14,23)(15,25)(16,24)(17,26)(18,28)(19,27)
(20,29)(21,31)(22,30)(33,34)(36,37)(39,40)(41,50)(42,52)(43,51)(44,53)(45,55)
(46,54)(47,56)(48,58)(49,57)(60,61)(63,64)(66,67)(68,77)(69,79)(70,78)(71,80)
(72,82)(73,81)(74,83)(75,85)(76,84);
s3 := Sym(85)!( 5,14)( 6,16)( 7,15)( 8,17)( 9,19)(10,18)(11,20)(12,22)(13,21)
(24,25)(27,28)(30,31)(32,41)(33,43)(34,42)(35,44)(36,46)(37,45)(38,47)(39,49)
(40,48)(51,52)(54,55)(57,58)(59,68)(60,70)(61,69)(62,71)(63,73)(64,72)(65,74)
(66,76)(67,75)(78,79)(81,82)(84,85);
s4 := Sym(85)!( 5,32)( 6,34)( 7,33)( 8,38)( 9,40)(10,39)(11,35)(12,37)(13,36)
(14,42)(15,41)(16,43)(17,48)(18,47)(19,49)(20,45)(21,44)(22,46)(23,52)(24,51)
(25,50)(26,58)(27,57)(28,56)(29,55)(30,54)(31,53)(59,62)(60,64)(61,63)(66,67)
(68,72)(69,71)(70,73)(74,75)(77,82)(78,81)(79,80)(83,85);
s5 := Sym(85)!( 6, 7)( 8,11)( 9,13)(10,12)(15,16)(17,20)(18,22)(19,21)(24,25)
(26,29)(27,31)(28,30)(32,62)(33,64)(34,63)(35,59)(36,61)(37,60)(38,65)(39,67)
(40,66)(41,71)(42,73)(43,72)(44,68)(45,70)(46,69)(47,74)(48,76)(49,75)(50,80)
(51,82)(52,81)(53,77)(54,79)(55,78)(56,83)(57,85)(58,84);
poly := sub<Sym(85)|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*s1*s0*s1, 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, 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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