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Polytope of Type {2,2,6,18}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,2,6,18}*1296c
if this polytope has a name.
Group : SmallGroup(1296,1862)
Rank : 5
Schlafli Type : {2,2,6,18}
Number of vertices, edges, etc : 2, 2, 9, 81, 27
Order of s0s1s2s3s4 : 6
Order of s0s1s2s3s4s3s2s1 : 2
Special Properties :
   Degenerate
   Universal
   Non-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 : {2,2,6,6}*432
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :
s0 := (1,2);;
s1 := (3,4);;
s2 := ( 8,12)( 9,13)(10,11)(14,23)(15,24)(16,25)(17,30)(18,31)(19,29)(20,28)
(21,26)(22,27)(35,39)(36,40)(37,38)(41,50)(42,51)(43,52)(44,57)(45,58)(46,56)
(47,55)(48,53)(49,54)(62,66)(63,67)(64,65)(68,77)(69,78)(70,79)(71,84)(72,85)
(73,83)(74,82)(75,80)(76,81);;
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)(32,70)(33,69)(34,68)(35,73)(36,72)(37,71)(38,76)(39,75)(40,74)
(41,61)(42,60)(43,59)(44,64)(45,63)(46,62)(47,67)(48,66)(49,65)(50,79)(51,78)
(52,77)(53,82)(54,81)(55,80)(56,85)(57,84)(58,83);;
s4 := ( 5,32)( 6,34)( 7,33)( 8,38)( 9,40)(10,39)(11,35)(12,37)(13,36)(14,44)
(15,46)(16,45)(17,41)(18,43)(19,42)(20,47)(21,49)(22,48)(23,57)(24,56)(25,58)
(26,54)(27,53)(28,55)(29,51)(30,50)(31,52)(59,61)(62,67)(63,66)(64,65)(68,73)
(69,72)(70,71)(74,76)(77,83)(78,85)(79,84)(81,82);;
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*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, s2*s3*s4*s2*s3*s4*s2*s3*s4, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s2*s3*s4*s3*s2*s3*s4*s3*s2*s3*s4*s3*s2*s3*s4*s3*s2*s3*s4*s3*s2*s3*s4*s3 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(85)!(1,2);
s1 := Sym(85)!(3,4);
s2 := Sym(85)!( 8,12)( 9,13)(10,11)(14,23)(15,24)(16,25)(17,30)(18,31)(19,29)
(20,28)(21,26)(22,27)(35,39)(36,40)(37,38)(41,50)(42,51)(43,52)(44,57)(45,58)
(46,56)(47,55)(48,53)(49,54)(62,66)(63,67)(64,65)(68,77)(69,78)(70,79)(71,84)
(72,85)(73,83)(74,82)(75,80)(76,81);
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)(32,70)(33,69)(34,68)(35,73)(36,72)(37,71)(38,76)(39,75)
(40,74)(41,61)(42,60)(43,59)(44,64)(45,63)(46,62)(47,67)(48,66)(49,65)(50,79)
(51,78)(52,77)(53,82)(54,81)(55,80)(56,85)(57,84)(58,83);
s4 := Sym(85)!( 5,32)( 6,34)( 7,33)( 8,38)( 9,40)(10,39)(11,35)(12,37)(13,36)
(14,44)(15,46)(16,45)(17,41)(18,43)(19,42)(20,47)(21,49)(22,48)(23,57)(24,56)
(25,58)(26,54)(27,53)(28,55)(29,51)(30,50)(31,52)(59,61)(62,67)(63,66)(64,65)
(68,73)(69,72)(70,71)(74,76)(77,83)(78,85)(79,84)(81,82);
poly := sub<Sym(85)|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*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, 
s2*s3*s4*s2*s3*s4*s2*s3*s4, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s2*s3*s4*s3*s2*s3*s4*s3*s2*s3*s4*s3*s2*s3*s4*s3*s2*s3*s4*s3*s2*s3*s4*s3 >; 
 

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