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Polytope of Type {2,14,6,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,14,6,4}*1344a
if this polytope has a name.
Group : SmallGroup(1344,11527)
Rank : 5
Schlafli Type : {2,14,6,4}
Number of vertices, edges, etc : 2, 14, 42, 12, 4
Order of s0s1s2s3s4 : 84
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 : {2,14,6,2}*672
   3-fold quotients : {2,14,2,4}*448
   6-fold quotients : {2,7,2,4}*224, {2,14,2,2}*224
   7-fold quotients : {2,2,6,4}*192a
   12-fold quotients : {2,7,2,2}*112
   14-fold quotients : {2,2,6,2}*96
   21-fold quotients : {2,2,2,4}*64
   28-fold quotients : {2,2,3,2}*48
   42-fold quotients : {2,2,2,2}*32
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :
s0 := (1,2);;
s1 := ( 4, 9)( 5, 8)( 6, 7)(11,16)(12,15)(13,14)(18,23)(19,22)(20,21)(25,30)
(26,29)(27,28)(32,37)(33,36)(34,35)(39,44)(40,43)(41,42)(46,51)(47,50)(48,49)
(53,58)(54,57)(55,56)(60,65)(61,64)(62,63)(67,72)(68,71)(69,70)(74,79)(75,78)
(76,77)(81,86)(82,85)(83,84);;
s2 := ( 3, 4)( 5, 9)( 6, 8)(10,18)(11,17)(12,23)(13,22)(14,21)(15,20)(16,19)
(24,25)(26,30)(27,29)(31,39)(32,38)(33,44)(34,43)(35,42)(36,41)(37,40)(45,46)
(47,51)(48,50)(52,60)(53,59)(54,65)(55,64)(56,63)(57,62)(58,61)(66,67)(68,72)
(69,71)(73,81)(74,80)(75,86)(76,85)(77,84)(78,83)(79,82);;
s3 := ( 3,10)( 4,11)( 5,12)( 6,13)( 7,14)( 8,15)( 9,16)(24,31)(25,32)(26,33)
(27,34)(28,35)(29,36)(30,37)(45,73)(46,74)(47,75)(48,76)(49,77)(50,78)(51,79)
(52,66)(53,67)(54,68)(55,69)(56,70)(57,71)(58,72)(59,80)(60,81)(61,82)(62,83)
(63,84)(64,85)(65,86);;
s4 := ( 3,45)( 4,46)( 5,47)( 6,48)( 7,49)( 8,50)( 9,51)(10,52)(11,53)(12,54)
(13,55)(14,56)(15,57)(16,58)(17,59)(18,60)(19,61)(20,62)(21,63)(22,64)(23,65)
(24,66)(25,67)(26,68)(27,69)(28,70)(29,71)(30,72)(31,73)(32,74)(33,75)(34,76)
(35,77)(36,78)(37,79)(38,80)(39,81)(40,82)(41,83)(42,84)(43,85)(44,86);;
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*s1*s0*s1, 
s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s0*s4*s0*s4, s1*s4*s1*s4, s2*s4*s2*s4, 
s1*s2*s3*s2*s1*s2*s3*s2, s2*s3*s4*s3*s2*s3*s4*s3, 
s3*s4*s3*s4*s3*s4*s3*s4, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
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)!(1,2);
s1 := Sym(86)!( 4, 9)( 5, 8)( 6, 7)(11,16)(12,15)(13,14)(18,23)(19,22)(20,21)
(25,30)(26,29)(27,28)(32,37)(33,36)(34,35)(39,44)(40,43)(41,42)(46,51)(47,50)
(48,49)(53,58)(54,57)(55,56)(60,65)(61,64)(62,63)(67,72)(68,71)(69,70)(74,79)
(75,78)(76,77)(81,86)(82,85)(83,84);
s2 := Sym(86)!( 3, 4)( 5, 9)( 6, 8)(10,18)(11,17)(12,23)(13,22)(14,21)(15,20)
(16,19)(24,25)(26,30)(27,29)(31,39)(32,38)(33,44)(34,43)(35,42)(36,41)(37,40)
(45,46)(47,51)(48,50)(52,60)(53,59)(54,65)(55,64)(56,63)(57,62)(58,61)(66,67)
(68,72)(69,71)(73,81)(74,80)(75,86)(76,85)(77,84)(78,83)(79,82);
s3 := Sym(86)!( 3,10)( 4,11)( 5,12)( 6,13)( 7,14)( 8,15)( 9,16)(24,31)(25,32)
(26,33)(27,34)(28,35)(29,36)(30,37)(45,73)(46,74)(47,75)(48,76)(49,77)(50,78)
(51,79)(52,66)(53,67)(54,68)(55,69)(56,70)(57,71)(58,72)(59,80)(60,81)(61,82)
(62,83)(63,84)(64,85)(65,86);
s4 := Sym(86)!( 3,45)( 4,46)( 5,47)( 6,48)( 7,49)( 8,50)( 9,51)(10,52)(11,53)
(12,54)(13,55)(14,56)(15,57)(16,58)(17,59)(18,60)(19,61)(20,62)(21,63)(22,64)
(23,65)(24,66)(25,67)(26,68)(27,69)(28,70)(29,71)(30,72)(31,73)(32,74)(33,75)
(34,76)(35,77)(36,78)(37,79)(38,80)(39,81)(40,82)(41,83)(42,84)(43,85)(44,86);
poly := sub<Sym(86)|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*s1*s0*s1, s0*s2*s0*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s0*s4*s0*s4, 
s1*s4*s1*s4, s2*s4*s2*s4, s1*s2*s3*s2*s1*s2*s3*s2, 
s2*s3*s4*s3*s2*s3*s4*s3, s3*s4*s3*s4*s3*s4*s3*s4, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
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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