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Polytope of Type {3,2,6,6,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,2,6,6,4}*1728c
if this polytope has a name.
Group : SmallGroup(1728,47409)
Rank : 6
Schlafli Type : {3,2,6,6,4}
Number of vertices, edges, etc : 3, 3, 6, 18, 12, 4
Order of s0s1s2s3s4s5 : 12
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) :
   2-fold quotients : {3,2,3,6,4}*864, {3,2,6,6,2}*864c
   3-fold quotients : {3,2,6,2,4}*576
   4-fold quotients : {3,2,3,6,2}*432
   6-fold quotients : {3,2,3,2,4}*288, {3,2,6,2,2}*288
   9-fold quotients : {3,2,2,2,4}*192
   12-fold quotients : {3,2,3,2,2}*144
   18-fold quotients : {3,2,2,2,2}*96
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :
s0 := (2,3);;
s1 := (1,2);;
s2 := ( 5, 6)( 7,10)( 8,12)( 9,11)(14,15)(16,19)(17,21)(18,20)(23,24)(25,28)
(26,30)(27,29)(32,33)(34,37)(35,39)(36,38)(41,42)(43,46)(44,48)(45,47)(50,51)
(52,55)(53,57)(54,56)(59,60)(61,64)(62,66)(63,65)(68,69)(70,73)(71,75)
(72,74);;
s3 := ( 4,44)( 5,43)( 6,45)( 7,41)( 8,40)( 9,42)(10,47)(11,46)(12,48)(13,53)
(14,52)(15,54)(16,50)(17,49)(18,51)(19,56)(20,55)(21,57)(22,62)(23,61)(24,63)
(25,59)(26,58)(27,60)(28,65)(29,64)(30,66)(31,71)(32,70)(33,72)(34,68)(35,67)
(36,69)(37,74)(38,73)(39,75);;
s4 := ( 7,10)( 8,11)( 9,12)(16,19)(17,20)(18,21)(22,31)(23,32)(24,33)(25,37)
(26,38)(27,39)(28,34)(29,35)(30,36)(43,46)(44,47)(45,48)(52,55)(53,56)(54,57)
(58,67)(59,68)(60,69)(61,73)(62,74)(63,75)(64,70)(65,71)(66,72);;
s5 := ( 4,22)( 5,23)( 6,24)( 7,25)( 8,26)( 9,27)(10,28)(11,29)(12,30)(13,31)
(14,32)(15,33)(16,34)(17,35)(18,36)(19,37)(20,38)(21,39)(40,58)(41,59)(42,60)
(43,61)(44,62)(45,63)(46,64)(47,65)(48,66)(49,67)(50,68)(51,69)(52,70)(53,71)
(54,72)(55,73)(56,74)(57,75);;
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*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, s0*s1*s0*s1*s0*s1, 
s3*s4*s5*s4*s3*s4*s5*s4, s4*s5*s4*s5*s4*s5*s4*s5, 
s4*s2*s3*s4*s3*s4*s2*s3*s4*s3, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s2*s3*s4*s3*s2*s3*s2*s3*s4*s3*s2*s3 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(75)!(2,3);
s1 := Sym(75)!(1,2);
s2 := Sym(75)!( 5, 6)( 7,10)( 8,12)( 9,11)(14,15)(16,19)(17,21)(18,20)(23,24)
(25,28)(26,30)(27,29)(32,33)(34,37)(35,39)(36,38)(41,42)(43,46)(44,48)(45,47)
(50,51)(52,55)(53,57)(54,56)(59,60)(61,64)(62,66)(63,65)(68,69)(70,73)(71,75)
(72,74);
s3 := Sym(75)!( 4,44)( 5,43)( 6,45)( 7,41)( 8,40)( 9,42)(10,47)(11,46)(12,48)
(13,53)(14,52)(15,54)(16,50)(17,49)(18,51)(19,56)(20,55)(21,57)(22,62)(23,61)
(24,63)(25,59)(26,58)(27,60)(28,65)(29,64)(30,66)(31,71)(32,70)(33,72)(34,68)
(35,67)(36,69)(37,74)(38,73)(39,75);
s4 := Sym(75)!( 7,10)( 8,11)( 9,12)(16,19)(17,20)(18,21)(22,31)(23,32)(24,33)
(25,37)(26,38)(27,39)(28,34)(29,35)(30,36)(43,46)(44,47)(45,48)(52,55)(53,56)
(54,57)(58,67)(59,68)(60,69)(61,73)(62,74)(63,75)(64,70)(65,71)(66,72);
s5 := Sym(75)!( 4,22)( 5,23)( 6,24)( 7,25)( 8,26)( 9,27)(10,28)(11,29)(12,30)
(13,31)(14,32)(15,33)(16,34)(17,35)(18,36)(19,37)(20,38)(21,39)(40,58)(41,59)
(42,60)(43,61)(44,62)(45,63)(46,64)(47,65)(48,66)(49,67)(50,68)(51,69)(52,70)
(53,71)(54,72)(55,73)(56,74)(57,75);
poly := sub<Sym(75)|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*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, 
s0*s1*s0*s1*s0*s1, s3*s4*s5*s4*s3*s4*s5*s4, 
s4*s5*s4*s5*s4*s5*s4*s5, s4*s2*s3*s4*s3*s4*s2*s3*s4*s3, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s2*s3*s4*s3*s2*s3*s2*s3*s4*s3*s2*s3 >; 
 

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