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Polytope of Type {3,2,18,8}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,2,18,8}*1728
if this polytope has a name.
Group : SmallGroup(1728,15957)
Rank : 5
Schlafli Type : {3,2,18,8}
Number of vertices, edges, etc : 3, 3, 18, 72, 8
Order of s0s1s2s3s4 : 72
Order of s0s1s2s3s4s3s2s1 : 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) :
   2-fold quotients : {3,2,18,4}*864a
   3-fold quotients : {3,2,6,8}*576
   4-fold quotients : {3,2,18,2}*432
   6-fold quotients : {3,2,6,4}*288a
   8-fold quotients : {3,2,9,2}*216
   9-fold quotients : {3,2,2,8}*192
   12-fold quotients : {3,2,6,2}*144
   18-fold quotients : {3,2,2,4}*96
   24-fold quotients : {3,2,3,2}*72
   36-fold quotients : {3,2,2,2}*48
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :
s0 := (2,3);;
s1 := (1,2);;
s2 := ( 5, 6)( 7,11)( 8,10)( 9,12)(14,15)(16,20)(17,19)(18,21)(23,24)(25,29)
(26,28)(27,30)(32,33)(34,38)(35,37)(36,39)(41,42)(43,47)(44,46)(45,48)(50,51)
(52,56)(53,55)(54,57)(59,60)(61,65)(62,64)(63,66)(68,69)(70,74)(71,73)
(72,75);;
s3 := ( 4, 7)( 5, 9)( 6, 8)(10,11)(13,16)(14,18)(15,17)(19,20)(22,34)(23,36)
(24,35)(25,31)(26,33)(27,32)(28,38)(29,37)(30,39)(40,61)(41,63)(42,62)(43,58)
(44,60)(45,59)(46,65)(47,64)(48,66)(49,70)(50,72)(51,71)(52,67)(53,69)(54,68)
(55,74)(56,73)(57,75);;
s4 := ( 4,40)( 5,41)( 6,42)( 7,43)( 8,44)( 9,45)(10,46)(11,47)(12,48)(13,49)
(14,50)(15,51)(16,52)(17,53)(18,54)(19,55)(20,56)(21,57)(22,67)(23,68)(24,69)
(25,70)(26,71)(27,72)(28,73)(29,74)(30,75)(31,58)(32,59)(33,60)(34,61)(35,62)
(36,63)(37,64)(38,65)(39,66);;
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*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*s1*s0*s1*s0*s1, s2*s3*s4*s3*s2*s3*s4*s3, 
s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(75)!(2,3);
s1 := Sym(75)!(1,2);
s2 := Sym(75)!( 5, 6)( 7,11)( 8,10)( 9,12)(14,15)(16,20)(17,19)(18,21)(23,24)
(25,29)(26,28)(27,30)(32,33)(34,38)(35,37)(36,39)(41,42)(43,47)(44,46)(45,48)
(50,51)(52,56)(53,55)(54,57)(59,60)(61,65)(62,64)(63,66)(68,69)(70,74)(71,73)
(72,75);
s3 := Sym(75)!( 4, 7)( 5, 9)( 6, 8)(10,11)(13,16)(14,18)(15,17)(19,20)(22,34)
(23,36)(24,35)(25,31)(26,33)(27,32)(28,38)(29,37)(30,39)(40,61)(41,63)(42,62)
(43,58)(44,60)(45,59)(46,65)(47,64)(48,66)(49,70)(50,72)(51,71)(52,67)(53,69)
(54,68)(55,74)(56,73)(57,75);
s4 := Sym(75)!( 4,40)( 5,41)( 6,42)( 7,43)( 8,44)( 9,45)(10,46)(11,47)(12,48)
(13,49)(14,50)(15,51)(16,52)(17,53)(18,54)(19,55)(20,56)(21,57)(22,67)(23,68)
(24,69)(25,70)(26,71)(27,72)(28,73)(29,74)(30,75)(31,58)(32,59)(33,60)(34,61)
(35,62)(36,63)(37,64)(38,65)(39,66);
poly := sub<Sym(75)|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*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*s1*s0*s1*s0*s1, 
s2*s3*s4*s3*s2*s3*s4*s3, s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >; 
 

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