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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}*1728b
if this polytope has a name.
Group : SmallGroup(1728,47319)
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}*864b, {6,2,6,6}*864b
   3-fold quotients : {2,2,6,12}*576b, {6,2,2,12}*576
   4-fold quotients : {3,2,6,6}*432b, {6,2,6,3}*432
   6-fold quotients : {3,2,2,12}*288, {2,2,6,6}*288b, {6,2,2,6}*288
   8-fold quotients : {3,2,6,3}*216
   9-fold quotients : {2,2,2,12}*192, {6,2,2,4}*192
   12-fold quotients : {2,2,6,3}*144, {3,2,2,6}*144, {6,2,2,3}*144
   18-fold quotients : {3,2,2,4}*96, {2,2,2,6}*96, {6,2,2,2}*96
   24-fold quotients : {3,2,2,3}*72
   27-fold quotients : {2,2,2,4}*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)(26,27)(29,30)(32,33)(35,36)
(38,39)(41,42)(44,45)(47,48)(50,51)(53,54)(56,57)(59,60)(62,63)(65,66)(68,69)
(71,72)(74,75)(77,78);;
s3 := ( 7,44)( 8,43)( 9,45)(10,50)(11,49)(12,51)(13,47)(14,46)(15,48)(16,53)
(17,52)(18,54)(19,59)(20,58)(21,60)(22,56)(23,55)(24,57)(25,71)(26,70)(27,72)
(28,77)(29,76)(30,78)(31,74)(32,73)(33,75)(34,62)(35,61)(36,63)(37,68)(38,67)
(39,69)(40,65)(41,64)(42,66);;
s4 := ( 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);;
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, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*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)(26,27)(29,30)(32,33)
(35,36)(38,39)(41,42)(44,45)(47,48)(50,51)(53,54)(56,57)(59,60)(62,63)(65,66)
(68,69)(71,72)(74,75)(77,78);
s3 := Sym(78)!( 7,44)( 8,43)( 9,45)(10,50)(11,49)(12,51)(13,47)(14,46)(15,48)
(16,53)(17,52)(18,54)(19,59)(20,58)(21,60)(22,56)(23,55)(24,57)(25,71)(26,70)
(27,72)(28,77)(29,76)(30,78)(31,74)(32,73)(33,75)(34,62)(35,61)(36,63)(37,68)
(38,67)(39,69)(40,65)(41,64)(42,66);
s4 := 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);
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, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4 >; 
 

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