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Polytope of Type {2,9,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,9,6}*1944a
if this polytope has a name.
Group : SmallGroup(1944,941)
Rank : 4
Schlafli Type : {2,9,6}
Number of vertices, edges, etc : 2, 81, 243, 54
Order of s0s1s2s3 : 6
Order of s0s1s2s3s2s1 : 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 : {2,9,6}*648b, {2,9,6}*648c
   9-fold quotients : {2,3,6}*216
   27-fold quotients : {2,3,6}*72
   81-fold quotients : {2,3,2}*24
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :
s0 := (1,2);;
s1 := ( 4, 5)( 6, 9)( 7,11)( 8,10)(12,20)(13,19)(14,18)(15,17)(21,25)(22,24)
(23,26)(27,28)(30,57)(31,59)(32,58)(33,63)(34,65)(35,64)(36,60)(37,62)(38,61)
(39,74)(40,73)(41,72)(42,71)(43,70)(44,69)(45,68)(46,67)(47,66)(48,79)(49,78)
(50,80)(51,76)(52,75)(53,77)(54,82)(55,81)(56,83);;
s2 := ( 3,39)( 4,41)( 5,40)( 6,45)( 7,47)( 8,46)( 9,42)(10,44)(11,43)(12,49)
(13,48)(14,50)(15,55)(16,54)(17,56)(18,52)(19,51)(20,53)(21,32)(22,31)(23,30)
(24,38)(25,37)(26,36)(27,35)(28,34)(29,33)(57,64)(58,63)(59,65)(60,61)(66,74)
(67,73)(68,72)(69,71)(75,81)(76,83)(77,82)(79,80);;
s3 := ( 4, 5)( 7, 8)(10,11)(12,24)(13,26)(14,25)(15,27)(16,29)(17,28)(18,21)
(19,23)(20,22)(30,57)(31,59)(32,58)(33,60)(34,62)(35,61)(36,63)(37,65)(38,64)
(39,78)(40,80)(41,79)(42,81)(43,83)(44,82)(45,75)(46,77)(47,76)(48,72)(49,74)
(50,73)(51,66)(52,68)(53,67)(54,69)(55,71)(56,70);;
poly := Group([s0,s1,s2,s3]);;
 
Finitely Presented Group Representation (GAP) :
F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s1*s0*s1, 
s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s3*s1*s2*s3*s2*s1*s2*s1*s2*s3*s1*s2*s3*s2*s1*s2*s1*s2, 
s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(83)!(1,2);
s1 := Sym(83)!( 4, 5)( 6, 9)( 7,11)( 8,10)(12,20)(13,19)(14,18)(15,17)(21,25)
(22,24)(23,26)(27,28)(30,57)(31,59)(32,58)(33,63)(34,65)(35,64)(36,60)(37,62)
(38,61)(39,74)(40,73)(41,72)(42,71)(43,70)(44,69)(45,68)(46,67)(47,66)(48,79)
(49,78)(50,80)(51,76)(52,75)(53,77)(54,82)(55,81)(56,83);
s2 := Sym(83)!( 3,39)( 4,41)( 5,40)( 6,45)( 7,47)( 8,46)( 9,42)(10,44)(11,43)
(12,49)(13,48)(14,50)(15,55)(16,54)(17,56)(18,52)(19,51)(20,53)(21,32)(22,31)
(23,30)(24,38)(25,37)(26,36)(27,35)(28,34)(29,33)(57,64)(58,63)(59,65)(60,61)
(66,74)(67,73)(68,72)(69,71)(75,81)(76,83)(77,82)(79,80);
s3 := Sym(83)!( 4, 5)( 7, 8)(10,11)(12,24)(13,26)(14,25)(15,27)(16,29)(17,28)
(18,21)(19,23)(20,22)(30,57)(31,59)(32,58)(33,60)(34,62)(35,61)(36,63)(37,65)
(38,64)(39,78)(40,80)(41,79)(42,81)(43,83)(44,82)(45,75)(46,77)(47,76)(48,72)
(49,74)(50,73)(51,66)(52,68)(53,67)(54,69)(55,71)(56,70);
poly := sub<Sym(83)|s0,s1,s2,s3>;
 
Finitely Presented Group Representation (Magma) :
poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2, 
s3*s3, s0*s1*s0*s1, s0*s2*s0*s2, s0*s3*s0*s3, 
s1*s3*s1*s3, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s3*s1*s2*s3*s2*s1*s2*s1*s2*s3*s1*s2*s3*s2*s1*s2*s1*s2, 
s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >; 
 

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