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

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,2,4,4,4}*768
if this polytope has a name.
Group : SmallGroup(768,336970)
Rank : 6
Schlafli Type : {3,2,4,4,4}
Number of vertices, edges, etc : 3, 3, 4, 8, 8, 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 : {3,2,2,4,4}*384, {3,2,4,4,2}*384, {3,2,4,2,4}*384
   4-fold quotients : {3,2,2,2,4}*192, {3,2,2,4,2}*192, {3,2,4,2,2}*192
   8-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 := ( 4,12)( 5,13)( 6,14)( 7,15)( 8,16)( 9,17)(10,18)(11,19)(20,28)(21,29)
(22,30)(23,31)(24,32)(25,33)(26,34)(27,35)(36,44)(37,45)(38,46)(39,47)(40,48)
(41,49)(42,50)(43,51)(52,60)(53,61)(54,62)(55,63)(56,64)(57,65)(58,66)
(59,67);;
s3 := (12,16)(13,17)(14,18)(15,19)(20,22)(21,23)(24,26)(25,27)(28,34)(29,35)
(30,32)(31,33)(36,40)(37,41)(38,42)(39,43)(52,58)(53,59)(54,56)(55,57)(60,62)
(61,63)(64,66)(65,67);;
s4 := ( 4,20)( 5,21)( 6,22)( 7,23)( 8,24)( 9,25)(10,26)(11,27)(12,28)(13,29)
(14,30)(15,31)(16,32)(17,33)(18,34)(19,35)(36,52)(37,53)(38,54)(39,55)(40,56)
(41,57)(42,58)(43,59)(44,60)(45,61)(46,62)(47,63)(48,64)(49,65)(50,66)
(51,67);;
s5 := ( 4,44)( 5,45)( 6,46)( 7,47)( 8,48)( 9,49)(10,50)(11,51)(12,36)(13,37)
(14,38)(15,39)(16,40)(17,41)(18,42)(19,43)(20,61)(21,60)(22,63)(23,62)(24,65)
(25,64)(26,67)(27,66)(28,53)(29,52)(30,55)(31,54)(32,57)(33,56)(34,59)
(35,58);;
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, 
s2*s3*s2*s3*s2*s3*s2*s3, s2*s3*s4*s3*s2*s3*s4*s3, 
s3*s4*s3*s4*s3*s4*s3*s4, s3*s4*s5*s4*s3*s4*s5*s4, 
s4*s5*s4*s5*s4*s5*s4*s5 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(67)!(2,3);
s1 := Sym(67)!(1,2);
s2 := Sym(67)!( 4,12)( 5,13)( 6,14)( 7,15)( 8,16)( 9,17)(10,18)(11,19)(20,28)
(21,29)(22,30)(23,31)(24,32)(25,33)(26,34)(27,35)(36,44)(37,45)(38,46)(39,47)
(40,48)(41,49)(42,50)(43,51)(52,60)(53,61)(54,62)(55,63)(56,64)(57,65)(58,66)
(59,67);
s3 := Sym(67)!(12,16)(13,17)(14,18)(15,19)(20,22)(21,23)(24,26)(25,27)(28,34)
(29,35)(30,32)(31,33)(36,40)(37,41)(38,42)(39,43)(52,58)(53,59)(54,56)(55,57)
(60,62)(61,63)(64,66)(65,67);
s4 := Sym(67)!( 4,20)( 5,21)( 6,22)( 7,23)( 8,24)( 9,25)(10,26)(11,27)(12,28)
(13,29)(14,30)(15,31)(16,32)(17,33)(18,34)(19,35)(36,52)(37,53)(38,54)(39,55)
(40,56)(41,57)(42,58)(43,59)(44,60)(45,61)(46,62)(47,63)(48,64)(49,65)(50,66)
(51,67);
s5 := Sym(67)!( 4,44)( 5,45)( 6,46)( 7,47)( 8,48)( 9,49)(10,50)(11,51)(12,36)
(13,37)(14,38)(15,39)(16,40)(17,41)(18,42)(19,43)(20,61)(21,60)(22,63)(23,62)
(24,65)(25,64)(26,67)(27,66)(28,53)(29,52)(30,55)(31,54)(32,57)(33,56)(34,59)
(35,58);
poly := sub<Sym(67)|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, s2*s3*s2*s3*s2*s3*s2*s3, 
s2*s3*s4*s3*s2*s3*s4*s3, s3*s4*s3*s4*s3*s4*s3*s4, 
s3*s4*s5*s4*s3*s4*s5*s4, s4*s5*s4*s5*s4*s5*s4*s5 >; 
 

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