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

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,2,6,12}*1728c
if this polytope has a name.
Group : SmallGroup(1728,47409)
Rank : 5
Schlafli Type : {6,2,6,12}
Number of vertices, edges, etc : 6, 6, 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 :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {3,2,6,12}*864c, {6,2,6,6}*864c
   3-fold quotients : {6,2,6,4}*576a, {2,2,6,12}*576c
   4-fold quotients : {3,2,6,6}*432c, {6,2,3,6}*432
   6-fold quotients : {3,2,6,4}*288a, {2,2,6,6}*288c, {6,2,6,2}*288
   8-fold quotients : {3,2,3,6}*216
   9-fold quotients : {2,2,6,4}*192a, {6,2,2,4}*192
   12-fold quotients : {2,2,3,6}*144, {3,2,6,2}*144, {6,2,3,2}*144
   18-fold quotients : {3,2,2,4}*96, {2,2,6,2}*96, {6,2,2,2}*96
   24-fold quotients : {3,2,3,2}*72
   27-fold quotients : {2,2,2,4}*64
   36-fold quotients : {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)( 8,45)( 9,44)(10,49)(11,51)(12,50)(13,46)(14,48)(15,47)(16,52)
(17,54)(18,53)(19,58)(20,60)(21,59)(22,55)(23,57)(24,56)(25,61)(26,63)(27,62)
(28,67)(29,69)(30,68)(31,64)(32,66)(33,65)(34,70)(35,72)(36,71)(37,76)(38,78)
(39,77)(40,73)(41,75)(42,74);;
s3 := ( 7,65)( 8,64)( 9,66)(10,62)(11,61)(12,63)(13,68)(14,67)(15,69)(16,74)
(17,73)(18,75)(19,71)(20,70)(21,72)(22,77)(23,76)(24,78)(25,47)(26,46)(27,48)
(28,44)(29,43)(30,45)(31,50)(32,49)(33,51)(34,56)(35,55)(36,57)(37,53)(38,52)
(39,54)(40,59)(41,58)(42,60);;
s4 := ( 8, 9)(11,12)(14,15)(17,18)(20,21)(23,24)(25,34)(26,36)(27,35)(28,37)
(29,39)(30,38)(31,40)(32,42)(33,41)(44,45)(47,48)(50,51)(53,54)(56,57)(59,60)
(61,70)(62,72)(63,71)(64,73)(65,75)(66,74)(67,76)(68,78)(69,77);;
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, 
s4*s2*s3*s4*s3*s4*s2*s3*s4*s3, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
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)( 8,45)( 9,44)(10,49)(11,51)(12,50)(13,46)(14,48)(15,47)
(16,52)(17,54)(18,53)(19,58)(20,60)(21,59)(22,55)(23,57)(24,56)(25,61)(26,63)
(27,62)(28,67)(29,69)(30,68)(31,64)(32,66)(33,65)(34,70)(35,72)(36,71)(37,76)
(38,78)(39,77)(40,73)(41,75)(42,74);
s3 := Sym(78)!( 7,65)( 8,64)( 9,66)(10,62)(11,61)(12,63)(13,68)(14,67)(15,69)
(16,74)(17,73)(18,75)(19,71)(20,70)(21,72)(22,77)(23,76)(24,78)(25,47)(26,46)
(27,48)(28,44)(29,43)(30,45)(31,50)(32,49)(33,51)(34,56)(35,55)(36,57)(37,53)
(38,52)(39,54)(40,59)(41,58)(42,60);
s4 := Sym(78)!( 8, 9)(11,12)(14,15)(17,18)(20,21)(23,24)(25,34)(26,36)(27,35)
(28,37)(29,39)(30,38)(31,40)(32,42)(33,41)(44,45)(47,48)(50,51)(53,54)(56,57)
(59,60)(61,70)(62,72)(63,71)(64,73)(65,75)(66,74)(67,76)(68,78)(69,77);
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, s4*s2*s3*s4*s3*s4*s2*s3*s4*s3, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >; 
 

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