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

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,3,6,9}*648
if this polytope has a name.
Group : SmallGroup(648,297)
Rank : 5
Schlafli Type : {2,3,6,9}
Number of vertices, edges, etc : 2, 3, 9, 27, 9
Order of s0s1s2s3s4 : 18
Order of s0s1s2s3s4s3s2s1 : 2
Special Properties :
   Degenerate
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {2,3,6,9,2} of size 1296
Vertex Figure Of :
   {2,2,3,6,9} of size 1296
   {3,2,3,6,9} of size 1944
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {2,3,2,9}*216, {2,3,6,3}*216
   9-fold quotients : {2,3,2,3}*72
Covers (Minimal Covers in Boldface) :
   2-fold covers : {2,3,6,18}*1296a, {2,6,6,9}*1296a
   3-fold covers : {2,9,6,9}*1944, {2,3,6,27}*1944, {2,3,6,9}*1944a, {2,3,6,9}*1944b, {6,3,6,9}*1944
Permutation Representation (GAP) :
s0 := (1,2);;
s1 := ( 4, 5)( 7, 8)(10,11)(12,21)(13,23)(14,22)(15,24)(16,26)(17,25)(18,27)
(19,29)(20,28)(31,32)(34,35)(37,38)(39,48)(40,50)(41,49)(42,51)(43,53)(44,52)
(45,54)(46,56)(47,55)(58,59)(61,62)(64,65)(66,75)(67,77)(68,76)(69,78)(70,80)
(71,79)(72,81)(73,83)(74,82);;
s2 := ( 3,12)( 4,14)( 5,13)( 6,15)( 7,17)( 8,16)( 9,18)(10,20)(11,19)(22,23)
(25,26)(28,29)(30,39)(31,41)(32,40)(33,42)(34,44)(35,43)(36,45)(37,47)(38,46)
(49,50)(52,53)(55,56)(57,66)(58,68)(59,67)(60,69)(61,71)(62,70)(63,72)(64,74)
(65,73)(76,77)(79,80)(82,83);;
s3 := ( 3,30)( 4,32)( 5,31)( 6,36)( 7,38)( 8,37)( 9,33)(10,35)(11,34)(12,40)
(13,39)(14,41)(15,46)(16,45)(17,47)(18,43)(19,42)(20,44)(21,50)(22,49)(23,48)
(24,56)(25,55)(26,54)(27,53)(28,52)(29,51)(57,60)(58,62)(59,61)(64,65)(66,70)
(67,69)(68,71)(72,73)(75,80)(76,79)(77,78)(81,83);;
s4 := ( 4, 5)( 6, 9)( 7,11)( 8,10)(13,14)(15,18)(16,20)(17,19)(22,23)(24,27)
(25,29)(26,28)(30,60)(31,62)(32,61)(33,57)(34,59)(35,58)(36,63)(37,65)(38,64)
(39,69)(40,71)(41,70)(42,66)(43,68)(44,67)(45,72)(46,74)(47,73)(48,78)(49,80)
(50,79)(51,75)(52,77)(53,76)(54,81)(55,83)(56,82);;
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, s0*s3*s0*s3, s1*s3*s1*s3, 
s0*s4*s0*s4, s1*s4*s1*s4, s2*s4*s2*s4, 
s1*s2*s1*s2*s1*s2, s3*s1*s2*s3*s2*s3*s1*s2*s3*s2, 
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(83)!(1,2);
s1 := Sym(83)!( 4, 5)( 7, 8)(10,11)(12,21)(13,23)(14,22)(15,24)(16,26)(17,25)
(18,27)(19,29)(20,28)(31,32)(34,35)(37,38)(39,48)(40,50)(41,49)(42,51)(43,53)
(44,52)(45,54)(46,56)(47,55)(58,59)(61,62)(64,65)(66,75)(67,77)(68,76)(69,78)
(70,80)(71,79)(72,81)(73,83)(74,82);
s2 := Sym(83)!( 3,12)( 4,14)( 5,13)( 6,15)( 7,17)( 8,16)( 9,18)(10,20)(11,19)
(22,23)(25,26)(28,29)(30,39)(31,41)(32,40)(33,42)(34,44)(35,43)(36,45)(37,47)
(38,46)(49,50)(52,53)(55,56)(57,66)(58,68)(59,67)(60,69)(61,71)(62,70)(63,72)
(64,74)(65,73)(76,77)(79,80)(82,83);
s3 := Sym(83)!( 3,30)( 4,32)( 5,31)( 6,36)( 7,38)( 8,37)( 9,33)(10,35)(11,34)
(12,40)(13,39)(14,41)(15,46)(16,45)(17,47)(18,43)(19,42)(20,44)(21,50)(22,49)
(23,48)(24,56)(25,55)(26,54)(27,53)(28,52)(29,51)(57,60)(58,62)(59,61)(64,65)
(66,70)(67,69)(68,71)(72,73)(75,80)(76,79)(77,78)(81,83);
s4 := Sym(83)!( 4, 5)( 6, 9)( 7,11)( 8,10)(13,14)(15,18)(16,20)(17,19)(22,23)
(24,27)(25,29)(26,28)(30,60)(31,62)(32,61)(33,57)(34,59)(35,58)(36,63)(37,65)
(38,64)(39,69)(40,71)(41,70)(42,66)(43,68)(44,67)(45,72)(46,74)(47,73)(48,78)
(49,80)(50,79)(51,75)(52,77)(53,76)(54,81)(55,83)(56,82);
poly := sub<Sym(83)|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, 
s0*s3*s0*s3, s1*s3*s1*s3, s0*s4*s0*s4, 
s1*s4*s1*s4, s2*s4*s2*s4, s1*s2*s1*s2*s1*s2, 
s3*s1*s2*s3*s2*s3*s1*s2*s3*s2, 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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