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Polytope of Type {2,42,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,42,4}*672a
if this polytope has a name.
Group : SmallGroup(672,1237)
Rank : 4
Schlafli Type : {2,42,4}
Number of vertices, edges, etc : 2, 42, 84, 4
Order of s0s1s2s3 : 84
Order of s0s1s2s3s2s1 : 2
Special Properties :
   Degenerate
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {2,42,4,2} of size 1344
Vertex Figure Of :
   {2,2,42,4} of size 1344
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {2,42,2}*336
   3-fold quotients : {2,14,4}*224
   4-fold quotients : {2,21,2}*168
   6-fold quotients : {2,14,2}*112
   7-fold quotients : {2,6,4}*96a
   12-fold quotients : {2,7,2}*56
   14-fold quotients : {2,6,2}*48
   21-fold quotients : {2,2,4}*32
   28-fold quotients : {2,3,2}*24
   42-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   2-fold covers : {2,84,4}*1344a, {4,42,4}*1344a, {2,42,8}*1344
Permutation Representation (GAP) :
s0 := (1,2);;
s1 := ( 4, 9)( 5, 8)( 6, 7)(10,17)(11,23)(12,22)(13,21)(14,20)(15,19)(16,18)
(25,30)(26,29)(27,28)(31,38)(32,44)(33,43)(34,42)(35,41)(36,40)(37,39)(46,51)
(47,50)(48,49)(52,59)(53,65)(54,64)(55,63)(56,62)(57,61)(58,60)(67,72)(68,71)
(69,70)(73,80)(74,86)(75,85)(76,84)(77,83)(78,82)(79,81);;
s2 := ( 3,11)( 4,10)( 5,16)( 6,15)( 7,14)( 8,13)( 9,12)(17,18)(19,23)(20,22)
(24,32)(25,31)(26,37)(27,36)(28,35)(29,34)(30,33)(38,39)(40,44)(41,43)(45,74)
(46,73)(47,79)(48,78)(49,77)(50,76)(51,75)(52,67)(53,66)(54,72)(55,71)(56,70)
(57,69)(58,68)(59,81)(60,80)(61,86)(62,85)(63,84)(64,83)(65,82);;
s3 := ( 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]);;
 
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*s1*s0*s1, 
s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s1*s2*s3*s2*s1*s2*s3*s2, 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*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)!(1,2);
s1 := Sym(86)!( 4, 9)( 5, 8)( 6, 7)(10,17)(11,23)(12,22)(13,21)(14,20)(15,19)
(16,18)(25,30)(26,29)(27,28)(31,38)(32,44)(33,43)(34,42)(35,41)(36,40)(37,39)
(46,51)(47,50)(48,49)(52,59)(53,65)(54,64)(55,63)(56,62)(57,61)(58,60)(67,72)
(68,71)(69,70)(73,80)(74,86)(75,85)(76,84)(77,83)(78,82)(79,81);
s2 := Sym(86)!( 3,11)( 4,10)( 5,16)( 6,15)( 7,14)( 8,13)( 9,12)(17,18)(19,23)
(20,22)(24,32)(25,31)(26,37)(27,36)(28,35)(29,34)(30,33)(38,39)(40,44)(41,43)
(45,74)(46,73)(47,79)(48,78)(49,77)(50,76)(51,75)(52,67)(53,66)(54,72)(55,71)
(56,70)(57,69)(58,68)(59,81)(60,80)(61,86)(62,85)(63,84)(64,83)(65,82);
s3 := 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>;
 
Finitely Presented Group Representation (Magma) :
poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2, 
s3*s3, s0*s1*s0*s1, s0*s2*s0*s2, s0*s3*s0*s3, 
s1*s3*s1*s3, s1*s2*s3*s2*s1*s2*s3*s2, 
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*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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