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

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,6,4,14}*1344
if this polytope has a name.
Group : SmallGroup(1344,11527)
Rank : 5
Schlafli Type : {2,6,4,14}
Number of vertices, edges, etc : 2, 6, 12, 28, 14
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 : {2,6,2,14}*672
   3-fold quotients : {2,2,4,14}*448
   4-fold quotients : {2,3,2,14}*336, {2,6,2,7}*336
   6-fold quotients : {2,2,2,14}*224
   7-fold quotients : {2,6,4,2}*192a
   8-fold quotients : {2,3,2,7}*168
   12-fold quotients : {2,2,2,7}*112
   14-fold quotients : {2,6,2,2}*96
   21-fold quotients : {2,2,4,2}*64
   28-fold quotients : {2,3,2,2}*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 := (10,17)(11,18)(12,19)(13,20)(14,21)(15,22)(16,23)(31,38)(32,39)(33,40)
(34,41)(35,42)(36,43)(37,44)(52,59)(53,60)(54,61)(55,62)(56,63)(57,64)(58,65)
(73,80)(74,81)(75,82)(76,83)(77,84)(78,85)(79,86);;
s2 := ( 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);;
s3 := ( 3,45)( 4,51)( 5,50)( 6,49)( 7,48)( 8,47)( 9,46)(10,52)(11,58)(12,57)
(13,56)(14,55)(15,54)(16,53)(17,59)(18,65)(19,64)(20,63)(21,62)(22,61)(23,60)
(24,66)(25,72)(26,71)(27,70)(28,69)(29,68)(30,67)(31,73)(32,79)(33,78)(34,77)
(35,76)(36,75)(37,74)(38,80)(39,86)(40,85)(41,84)(42,83)(43,82)(44,81);;
s4 := ( 3, 4)( 5, 9)( 6, 8)(10,11)(12,16)(13,15)(17,18)(19,23)(20,22)(24,25)
(26,30)(27,29)(31,32)(33,37)(34,36)(38,39)(40,44)(41,43)(45,46)(47,51)(48,50)
(52,53)(54,58)(55,57)(59,60)(61,65)(62,64)(66,67)(68,72)(69,71)(73,74)(75,79)
(76,78)(80,81)(82,86)(83,85);;
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*s2*s3*s2*s3*s2*s3, 
s2*s3*s4*s3*s2*s3*s4*s3, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s3*s4*s3*s4*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(86)!(1,2);
s1 := Sym(86)!(10,17)(11,18)(12,19)(13,20)(14,21)(15,22)(16,23)(31,38)(32,39)
(33,40)(34,41)(35,42)(36,43)(37,44)(52,59)(53,60)(54,61)(55,62)(56,63)(57,64)
(58,65)(73,80)(74,81)(75,82)(76,83)(77,84)(78,85)(79,86);
s2 := 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);
s3 := Sym(86)!( 3,45)( 4,51)( 5,50)( 6,49)( 7,48)( 8,47)( 9,46)(10,52)(11,58)
(12,57)(13,56)(14,55)(15,54)(16,53)(17,59)(18,65)(19,64)(20,63)(21,62)(22,61)
(23,60)(24,66)(25,72)(26,71)(27,70)(28,69)(29,68)(30,67)(31,73)(32,79)(33,78)
(34,77)(35,76)(36,75)(37,74)(38,80)(39,86)(40,85)(41,84)(42,83)(43,82)(44,81);
s4 := Sym(86)!( 3, 4)( 5, 9)( 6, 8)(10,11)(12,16)(13,15)(17,18)(19,23)(20,22)
(24,25)(26,30)(27,29)(31,32)(33,37)(34,36)(38,39)(40,44)(41,43)(45,46)(47,51)
(48,50)(52,53)(54,58)(55,57)(59,60)(61,65)(62,64)(66,67)(68,72)(69,71)(73,74)
(75,79)(76,78)(80,81)(82,86)(83,85);
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*s2*s3*s2*s3*s2*s3, s2*s3*s4*s3*s2*s3*s4*s3, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s3*s4*s3*s4*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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