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Polytope of Type {14,4,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {14,4,6}*672
Also Known As : {{14,4|2},{4,6|2}}. if this polytope has another name.
Group : SmallGroup(672,1150)
Rank : 4
Schlafli Type : {14,4,6}
Number of vertices, edges, etc : 14, 28, 12, 6
Order of s0s1s2s3 : 84
Order of s0s1s2s3s2s1 : 2
Special Properties :
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {14,4,6,2} of size 1344
Vertex Figure Of :
   {2,14,4,6} of size 1344
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {14,2,6}*336
   3-fold quotients : {14,4,2}*224
   4-fold quotients : {7,2,6}*168, {14,2,3}*168
   6-fold quotients : {14,2,2}*112
   7-fold quotients : {2,4,6}*96a
   8-fold quotients : {7,2,3}*84
   12-fold quotients : {7,2,2}*56
   14-fold quotients : {2,2,6}*48
   21-fold quotients : {2,4,2}*32
   28-fold quotients : {2,2,3}*24
   42-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   2-fold covers : {14,4,12}*1344, {28,4,6}*1344, {14,8,6}*1344
Permutation Representation (GAP) :
s0 := ( 2, 7)( 3, 6)( 4, 5)( 9,14)(10,13)(11,12)(16,21)(17,20)(18,19)(23,28)
(24,27)(25,26)(30,35)(31,34)(32,33)(37,42)(38,41)(39,40)(44,49)(45,48)(46,47)
(51,56)(52,55)(53,54)(58,63)(59,62)(60,61)(65,70)(66,69)(67,68)(72,77)(73,76)
(74,75)(79,84)(80,83)(81,82);;
s1 := ( 1, 2)( 3, 7)( 4, 6)( 8, 9)(10,14)(11,13)(15,16)(17,21)(18,20)(22,23)
(24,28)(25,27)(29,30)(31,35)(32,34)(36,37)(38,42)(39,41)(43,65)(44,64)(45,70)
(46,69)(47,68)(48,67)(49,66)(50,72)(51,71)(52,77)(53,76)(54,75)(55,74)(56,73)
(57,79)(58,78)(59,84)(60,83)(61,82)(62,81)(63,80);;
s2 := ( 1,43)( 2,44)( 3,45)( 4,46)( 5,47)( 6,48)( 7,49)( 8,57)( 9,58)(10,59)
(11,60)(12,61)(13,62)(14,63)(15,50)(16,51)(17,52)(18,53)(19,54)(20,55)(21,56)
(22,64)(23,65)(24,66)(25,67)(26,68)(27,69)(28,70)(29,78)(30,79)(31,80)(32,81)
(33,82)(34,83)(35,84)(36,71)(37,72)(38,73)(39,74)(40,75)(41,76)(42,77);;
s3 := ( 1, 8)( 2, 9)( 3,10)( 4,11)( 5,12)( 6,13)( 7,14)(22,29)(23,30)(24,31)
(25,32)(26,33)(27,34)(28,35)(43,50)(44,51)(45,52)(46,53)(47,54)(48,55)(49,56)
(64,71)(65,72)(66,73)(67,74)(68,75)(69,76)(70,77);;
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, s0*s1*s2*s1*s0*s1*s2*s1, 
s1*s2*s1*s2*s1*s2*s1*s2, s1*s2*s3*s2*s1*s2*s3*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(84)!( 2, 7)( 3, 6)( 4, 5)( 9,14)(10,13)(11,12)(16,21)(17,20)(18,19)
(23,28)(24,27)(25,26)(30,35)(31,34)(32,33)(37,42)(38,41)(39,40)(44,49)(45,48)
(46,47)(51,56)(52,55)(53,54)(58,63)(59,62)(60,61)(65,70)(66,69)(67,68)(72,77)
(73,76)(74,75)(79,84)(80,83)(81,82);
s1 := Sym(84)!( 1, 2)( 3, 7)( 4, 6)( 8, 9)(10,14)(11,13)(15,16)(17,21)(18,20)
(22,23)(24,28)(25,27)(29,30)(31,35)(32,34)(36,37)(38,42)(39,41)(43,65)(44,64)
(45,70)(46,69)(47,68)(48,67)(49,66)(50,72)(51,71)(52,77)(53,76)(54,75)(55,74)
(56,73)(57,79)(58,78)(59,84)(60,83)(61,82)(62,81)(63,80);
s2 := Sym(84)!( 1,43)( 2,44)( 3,45)( 4,46)( 5,47)( 6,48)( 7,49)( 8,57)( 9,58)
(10,59)(11,60)(12,61)(13,62)(14,63)(15,50)(16,51)(17,52)(18,53)(19,54)(20,55)
(21,56)(22,64)(23,65)(24,66)(25,67)(26,68)(27,69)(28,70)(29,78)(30,79)(31,80)
(32,81)(33,82)(34,83)(35,84)(36,71)(37,72)(38,73)(39,74)(40,75)(41,76)(42,77);
s3 := Sym(84)!( 1, 8)( 2, 9)( 3,10)( 4,11)( 5,12)( 6,13)( 7,14)(22,29)(23,30)
(24,31)(25,32)(26,33)(27,34)(28,35)(43,50)(44,51)(45,52)(46,53)(47,54)(48,55)
(49,56)(64,71)(65,72)(66,73)(67,74)(68,75)(69,76)(70,77);
poly := sub<Sym(84)|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, 
s0*s1*s2*s1*s0*s1*s2*s1, s1*s2*s1*s2*s1*s2*s1*s2, 
s1*s2*s3*s2*s1*s2*s3*s2, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >; 
 
References : None.
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