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

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,2,6,12}*864c
if this polytope has a name.
Group : SmallGroup(864,4406)
Rank : 5
Schlafli Type : {3,2,6,12}
Number of vertices, edges, etc : 3, 3, 6, 36, 12
Order of s0s1s2s3s4 : 12
Order of s0s1s2s3s4s3s2s1 : 2
Special Properties :
   Degenerate
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {3,2,6,12,2} of size 1728
Vertex Figure Of :
   {2,3,2,6,12} of size 1728
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {3,2,6,6}*432c
   3-fold quotients : {3,2,6,4}*288a
   4-fold quotients : {3,2,3,6}*216
   6-fold quotients : {3,2,6,2}*144
   9-fold quotients : {3,2,2,4}*96
   12-fold quotients : {3,2,3,2}*72
   18-fold quotients : {3,2,2,2}*48
Covers (Minimal Covers in Boldface) :
   2-fold covers : {3,2,12,12}*1728c, {3,2,6,24}*1728c, {6,2,6,12}*1728c
Permutation Representation (GAP) :
s0 := (2,3);;
s1 := (1,2);;
s2 := ( 4,40)( 5,42)( 6,41)( 7,46)( 8,48)( 9,47)(10,43)(11,45)(12,44)(13,49)
(14,51)(15,50)(16,55)(17,57)(18,56)(19,52)(20,54)(21,53)(22,58)(23,60)(24,59)
(25,64)(26,66)(27,65)(28,61)(29,63)(30,62)(31,67)(32,69)(33,68)(34,73)(35,75)
(36,74)(37,70)(38,72)(39,71);;
s3 := ( 4,62)( 5,61)( 6,63)( 7,59)( 8,58)( 9,60)(10,65)(11,64)(12,66)(13,71)
(14,70)(15,72)(16,68)(17,67)(18,69)(19,74)(20,73)(21,75)(22,44)(23,43)(24,45)
(25,41)(26,40)(27,42)(28,47)(29,46)(30,48)(31,53)(32,52)(33,54)(34,50)(35,49)
(36,51)(37,56)(38,55)(39,57);;
s4 := ( 5, 6)( 8, 9)(11,12)(14,15)(17,18)(20,21)(22,31)(23,33)(24,32)(25,34)
(26,36)(27,35)(28,37)(29,39)(30,38)(41,42)(44,45)(47,48)(50,51)(53,54)(56,57)
(58,67)(59,69)(60,68)(61,70)(62,72)(63,71)(64,73)(65,75)(66,74);;
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, s4*s2*s3*s4*s3*s4*s2*s3*s4*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)!( 4,40)( 5,42)( 6,41)( 7,46)( 8,48)( 9,47)(10,43)(11,45)(12,44)
(13,49)(14,51)(15,50)(16,55)(17,57)(18,56)(19,52)(20,54)(21,53)(22,58)(23,60)
(24,59)(25,64)(26,66)(27,65)(28,61)(29,63)(30,62)(31,67)(32,69)(33,68)(34,73)
(35,75)(36,74)(37,70)(38,72)(39,71);
s3 := Sym(75)!( 4,62)( 5,61)( 6,63)( 7,59)( 8,58)( 9,60)(10,65)(11,64)(12,66)
(13,71)(14,70)(15,72)(16,68)(17,67)(18,69)(19,74)(20,73)(21,75)(22,44)(23,43)
(24,45)(25,41)(26,40)(27,42)(28,47)(29,46)(30,48)(31,53)(32,52)(33,54)(34,50)
(35,49)(36,51)(37,56)(38,55)(39,57);
s4 := Sym(75)!( 5, 6)( 8, 9)(11,12)(14,15)(17,18)(20,21)(22,31)(23,33)(24,32)
(25,34)(26,36)(27,35)(28,37)(29,39)(30,38)(41,42)(44,45)(47,48)(50,51)(53,54)
(56,57)(58,67)(59,69)(60,68)(61,70)(62,72)(63,71)(64,73)(65,75)(66,74);
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, 
s4*s2*s3*s4*s3*s4*s2*s3*s4*s3, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >; 
 

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