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Polytope of Type {3,2,12,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,2,12,6}*1728b
if this polytope has a name.
Group : SmallGroup(1728,47874)
Rank : 5
Schlafli Type : {3,2,12,6}
Number of vertices, edges, etc : 3, 3, 24, 72, 12
Order of s0s1s2s3s4 : 6
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 : {3,2,12,3}*864
   3-fold quotients : {3,2,4,6}*576
   4-fold quotients : {3,2,6,6}*432b
   6-fold quotients : {3,2,4,3}*288, {3,2,4,6}*288b, {3,2,4,6}*288c
   8-fold quotients : {3,2,6,3}*216
   12-fold quotients : {3,2,4,3}*144, {3,2,2,6}*144
   24-fold quotients : {3,2,2,3}*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 := ( 4, 5)( 6, 7)( 8,13)( 9,12)(10,15)(11,14)(16,17)(18,19)(20,25)(21,24)
(22,27)(23,26)(28,29)(30,31)(32,37)(33,36)(34,39)(35,38)(40,41)(42,43)(44,49)
(45,48)(46,51)(47,50)(52,53)(54,55)(56,61)(57,60)(58,63)(59,62)(64,65)(66,67)
(68,73)(69,72)(70,75)(71,74);;
s3 := ( 4, 8)( 5,10)( 6, 9)( 7,11)(13,14)(16,32)(17,34)(18,33)(19,35)(20,28)
(21,30)(22,29)(23,31)(24,36)(25,38)(26,37)(27,39)(40,44)(41,46)(42,45)(43,47)
(49,50)(52,68)(53,70)(54,69)(55,71)(56,64)(57,66)(58,65)(59,67)(60,72)(61,74)
(62,73)(63,75);;
s4 := ( 4,52)( 5,53)( 6,55)( 7,54)( 8,60)( 9,61)(10,63)(11,62)(12,56)(13,57)
(14,59)(15,58)(16,40)(17,41)(18,43)(19,42)(20,48)(21,49)(22,51)(23,50)(24,44)
(25,45)(26,47)(27,46)(28,64)(29,65)(30,67)(31,66)(32,72)(33,73)(34,75)(35,74)
(36,68)(37,69)(38,71)(39,70);;
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*s4*s3*s2*s3*s4*s3*s4*s3, 
s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4, 
s4*s2*s3*s2*s3*s2*s3*s2*s3*s4*s2*s3*s2*s3*s2*s3*s2*s3, 
s2*s3*s4*s3*s2*s3*s2*s3*s4*s3*s2*s3*s4*s2*s3*s2*s4*s3 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(75)!(2,3);
s1 := Sym(75)!(1,2);
s2 := Sym(75)!( 4, 5)( 6, 7)( 8,13)( 9,12)(10,15)(11,14)(16,17)(18,19)(20,25)
(21,24)(22,27)(23,26)(28,29)(30,31)(32,37)(33,36)(34,39)(35,38)(40,41)(42,43)
(44,49)(45,48)(46,51)(47,50)(52,53)(54,55)(56,61)(57,60)(58,63)(59,62)(64,65)
(66,67)(68,73)(69,72)(70,75)(71,74);
s3 := Sym(75)!( 4, 8)( 5,10)( 6, 9)( 7,11)(13,14)(16,32)(17,34)(18,33)(19,35)
(20,28)(21,30)(22,29)(23,31)(24,36)(25,38)(26,37)(27,39)(40,44)(41,46)(42,45)
(43,47)(49,50)(52,68)(53,70)(54,69)(55,71)(56,64)(57,66)(58,65)(59,67)(60,72)
(61,74)(62,73)(63,75);
s4 := Sym(75)!( 4,52)( 5,53)( 6,55)( 7,54)( 8,60)( 9,61)(10,63)(11,62)(12,56)
(13,57)(14,59)(15,58)(16,40)(17,41)(18,43)(19,42)(20,48)(21,49)(22,51)(23,50)
(24,44)(25,45)(26,47)(27,46)(28,64)(29,65)(30,67)(31,66)(32,72)(33,73)(34,75)
(35,74)(36,68)(37,69)(38,71)(39,70);
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*s4*s3*s2*s3*s4*s3*s4*s3, 
s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4, 
s4*s2*s3*s2*s3*s2*s3*s2*s3*s4*s2*s3*s2*s3*s2*s3*s2*s3, 
s2*s3*s4*s3*s2*s3*s2*s3*s4*s3*s2*s3*s4*s2*s3*s2*s4*s3 >; 
 

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