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

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,2,12,6}*1728a
if this polytope has a name.
Group : SmallGroup(1728,47319)
Rank : 5
Schlafli Type : {6,2,12,6}
Number of vertices, edges, etc : 6, 6, 12, 36, 6
Order of s0s1s2s3s4 : 12
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 : {3,2,12,6}*864a, {6,2,6,6}*864a
   3-fold quotients : {2,2,12,6}*576a, {6,2,12,2}*576, {6,2,4,6}*576a
   4-fold quotients : {3,2,6,6}*432a
   6-fold quotients : {3,2,12,2}*288, {3,2,4,6}*288a, {2,2,6,6}*288a, {6,2,2,6}*288, {6,2,6,2}*288
   9-fold quotients : {2,2,12,2}*192, {2,2,4,6}*192a, {6,2,4,2}*192
   12-fold quotients : {3,2,2,6}*144, {3,2,6,2}*144, {6,2,2,3}*144, {6,2,3,2}*144
   18-fold quotients : {3,2,4,2}*96, {2,2,2,6}*96, {2,2,6,2}*96, {6,2,2,2}*96
   24-fold quotients : {3,2,2,3}*72, {3,2,3,2}*72
   27-fold quotients : {2,2,4,2}*64
   36-fold quotients : {2,2,2,3}*48, {2,2,3,2}*48, {3,2,2,2}*48
   54-fold quotients : {2,2,2,2}*32
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :
s0 := (3,4)(5,6);;
s1 := (1,5)(2,3)(4,6);;
s2 := ( 7,43)( 8,44)( 9,45)(10,49)(11,50)(12,51)(13,46)(14,47)(15,48)(16,52)
(17,53)(18,54)(19,58)(20,59)(21,60)(22,55)(23,56)(24,57)(25,70)(26,71)(27,72)
(28,76)(29,77)(30,78)(31,73)(32,74)(33,75)(34,61)(35,62)(36,63)(37,67)(38,68)
(39,69)(40,64)(41,65)(42,66);;
s3 := ( 7,64)( 8,66)( 9,65)(10,61)(11,63)(12,62)(13,67)(14,69)(15,68)(16,73)
(17,75)(18,74)(19,70)(20,72)(21,71)(22,76)(23,78)(24,77)(25,46)(26,48)(27,47)
(28,43)(29,45)(30,44)(31,49)(32,51)(33,50)(34,55)(35,57)(36,56)(37,52)(38,54)
(39,53)(40,58)(41,60)(42,59);;
s4 := ( 7, 8)(10,11)(13,14)(16,17)(19,20)(22,23)(25,26)(28,29)(31,32)(34,35)
(37,38)(40,41)(43,44)(46,47)(49,50)(52,53)(55,56)(58,59)(61,62)(64,65)(67,68)
(70,71)(73,74)(76,77);;
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, 
s2*s3*s4*s3*s2*s3*s4*s3, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
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 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(78)!(3,4)(5,6);
s1 := Sym(78)!(1,5)(2,3)(4,6);
s2 := Sym(78)!( 7,43)( 8,44)( 9,45)(10,49)(11,50)(12,51)(13,46)(14,47)(15,48)
(16,52)(17,53)(18,54)(19,58)(20,59)(21,60)(22,55)(23,56)(24,57)(25,70)(26,71)
(27,72)(28,76)(29,77)(30,78)(31,73)(32,74)(33,75)(34,61)(35,62)(36,63)(37,67)
(38,68)(39,69)(40,64)(41,65)(42,66);
s3 := Sym(78)!( 7,64)( 8,66)( 9,65)(10,61)(11,63)(12,62)(13,67)(14,69)(15,68)
(16,73)(17,75)(18,74)(19,70)(20,72)(21,71)(22,76)(23,78)(24,77)(25,46)(26,48)
(27,47)(28,43)(29,45)(30,44)(31,49)(32,51)(33,50)(34,55)(35,57)(36,56)(37,52)
(38,54)(39,53)(40,58)(41,60)(42,59);
s4 := Sym(78)!( 7, 8)(10,11)(13,14)(16,17)(19,20)(22,23)(25,26)(28,29)(31,32)
(34,35)(37,38)(40,41)(43,44)(46,47)(49,50)(52,53)(55,56)(58,59)(61,62)(64,65)
(67,68)(70,71)(73,74)(76,77);
poly := sub<Sym(78)|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, s2*s3*s4*s3*s2*s3*s4*s3, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
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 >; 
 

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