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

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

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