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Polytope of Type {6,28,2}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,28,2}*672a
if this polytope has a name.
Group : SmallGroup(672,1141)
Rank : 4
Schlafli Type : {6,28,2}
Number of vertices, edges, etc : 6, 84, 28, 2
Order of s0s1s2s3 : 84
Order of s0s1s2s3s2s1 : 2
Special Properties :
   Degenerate
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {6,28,2,2} of size 1344
Vertex Figure Of :
   {2,6,28,2} of size 1344
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {6,14,2}*336
   3-fold quotients : {2,28,2}*224
   6-fold quotients : {2,14,2}*112
   7-fold quotients : {6,4,2}*96a
   12-fold quotients : {2,7,2}*56
   14-fold quotients : {6,2,2}*48
   21-fold quotients : {2,4,2}*32
   28-fold quotients : {3,2,2}*24
   42-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   2-fold covers : {6,28,4}*1344, {6,56,2}*1344, {12,28,2}*1344
Permutation Representation (GAP) :
s0 := ( 8,15)( 9,16)(10,17)(11,18)(12,19)(13,20)(14,21)(29,36)(30,37)(31,38)
(32,39)(33,40)(34,41)(35,42)(50,57)(51,58)(52,59)(53,60)(54,61)(55,62)(56,63)
(71,78)(72,79)(73,80)(74,81)(75,82)(76,83)(77,84);;
s1 := ( 1, 8)( 2,14)( 3,13)( 4,12)( 5,11)( 6,10)( 7, 9)(16,21)(17,20)(18,19)
(22,29)(23,35)(24,34)(25,33)(26,32)(27,31)(28,30)(37,42)(38,41)(39,40)(43,71)
(44,77)(45,76)(46,75)(47,74)(48,73)(49,72)(50,64)(51,70)(52,69)(53,68)(54,67)
(55,66)(56,65)(57,78)(58,84)(59,83)(60,82)(61,81)(62,80)(63,79);;
s2 := ( 1,44)( 2,43)( 3,49)( 4,48)( 5,47)( 6,46)( 7,45)( 8,51)( 9,50)(10,56)
(11,55)(12,54)(13,53)(14,52)(15,58)(16,57)(17,63)(18,62)(19,61)(20,60)(21,59)
(22,65)(23,64)(24,70)(25,69)(26,68)(27,67)(28,66)(29,72)(30,71)(31,77)(32,76)
(33,75)(34,74)(35,73)(36,79)(37,78)(38,84)(39,83)(40,82)(41,81)(42,80);;
s3 := (85,86);;
poly := Group([s0,s1,s2,s3]);;
 
Finitely Presented Group Representation (GAP) :
F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s2*s0*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s2*s3*s2*s3, 
s0*s1*s2*s1*s0*s1*s2*s1, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(86)!( 8,15)( 9,16)(10,17)(11,18)(12,19)(13,20)(14,21)(29,36)(30,37)
(31,38)(32,39)(33,40)(34,41)(35,42)(50,57)(51,58)(52,59)(53,60)(54,61)(55,62)
(56,63)(71,78)(72,79)(73,80)(74,81)(75,82)(76,83)(77,84);
s1 := Sym(86)!( 1, 8)( 2,14)( 3,13)( 4,12)( 5,11)( 6,10)( 7, 9)(16,21)(17,20)
(18,19)(22,29)(23,35)(24,34)(25,33)(26,32)(27,31)(28,30)(37,42)(38,41)(39,40)
(43,71)(44,77)(45,76)(46,75)(47,74)(48,73)(49,72)(50,64)(51,70)(52,69)(53,68)
(54,67)(55,66)(56,65)(57,78)(58,84)(59,83)(60,82)(61,81)(62,80)(63,79);
s2 := Sym(86)!( 1,44)( 2,43)( 3,49)( 4,48)( 5,47)( 6,46)( 7,45)( 8,51)( 9,50)
(10,56)(11,55)(12,54)(13,53)(14,52)(15,58)(16,57)(17,63)(18,62)(19,61)(20,60)
(21,59)(22,65)(23,64)(24,70)(25,69)(26,68)(27,67)(28,66)(29,72)(30,71)(31,77)
(32,76)(33,75)(34,74)(35,73)(36,79)(37,78)(38,84)(39,83)(40,82)(41,81)(42,80);
s3 := Sym(86)!(85,86);
poly := sub<Sym(86)|s0,s1,s2,s3>;
 
Finitely Presented Group Representation (Magma) :
poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2, 
s3*s3, s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s2*s3*s2*s3, s0*s1*s2*s1*s0*s1*s2*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >; 
 

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