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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}*1728c
if this polytope has a name.
Group : SmallGroup(1728,47409)
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}*864c, {6,2,6,6}*864b
   3-fold quotients : {6,2,4,6}*576a, {2,2,12,6}*576c
   4-fold quotients : {3,2,6,6}*432b, {6,2,6,3}*432
   6-fold quotients : {3,2,4,6}*288a, {2,2,6,6}*288b, {6,2,2,6}*288
   8-fold quotients : {3,2,6,3}*216
   9-fold quotients : {2,2,4,6}*192a, {6,2,4,2}*192
   12-fold quotients : {2,2,6,3}*144, {3,2,2,6}*144, {6,2,2,3}*144
   18-fold quotients : {3,2,4,2}*96, {2,2,2,6}*96, {6,2,2,2}*96
   24-fold quotients : {3,2,2,3}*72
   27-fold quotients : {2,2,4,2}*64
   36-fold quotients : {2,2,2,3}*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 := ( 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);;
s3 := ( 7,62)( 8,61)( 9,63)(10,68)(11,67)(12,69)(13,65)(14,64)(15,66)(16,71)
(17,70)(18,72)(19,77)(20,76)(21,78)(22,74)(23,73)(24,75)(25,44)(26,43)(27,45)
(28,50)(29,49)(30,51)(31,47)(32,46)(33,48)(34,53)(35,52)(36,54)(37,59)(38,58)
(39,60)(40,56)(41,55)(42,57);;
s4 := ( 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)(40,76)(41,78)(42,77);;
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*s2*s3*s4*s2*s3*s2*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 ];;
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)!( 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);
s3 := Sym(78)!( 7,62)( 8,61)( 9,63)(10,68)(11,67)(12,69)(13,65)(14,64)(15,66)
(16,71)(17,70)(18,72)(19,77)(20,76)(21,78)(22,74)(23,73)(24,75)(25,44)(26,43)
(27,45)(28,50)(29,49)(30,51)(31,47)(32,46)(33,48)(34,53)(35,52)(36,54)(37,59)
(38,58)(39,60)(40,56)(41,55)(42,57);
s4 := Sym(78)!( 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)(40,76)(41,78)(42,77);
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*s2*s3*s4*s2*s3*s2*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 >; 
 

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