Polytope of Type {14,6,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {14,6,4}*672a
Also Known As : {{14,6|2},{6,4|2}}. if this polytope has another name.
Group : SmallGroup(672,1150)
Rank : 4
Schlafli Type : {14,6,4}
Number of vertices, edges, etc : 14, 42, 12, 4
Order of s₀s₁s₂s₃ : 84
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {14,6,4,2} of size 1344
Vertex Figure Of :
   {2,14,6,4} of size 1344
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {14,6,2}*336
   3-fold quotients : {14,2,4}*224
   6-fold quotients : {7,2,4}*112, {14,2,2}*112
   7-fold quotients : {2,6,4}*96a
   12-fold quotients : {7,2,2}*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 : {28,6,4}*1344a, {14,12,4}*1344a, {14,6,8}*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)( 8,16)( 9,15)(10,21)(11,20)(12,19)(13,18)(14,17)
(22,23)(24,28)(25,27)(29,37)(30,36)(31,42)(32,41)(33,40)(34,39)(35,38)(43,44)
(45,49)(46,48)(50,58)(51,57)(52,63)(53,62)(54,61)(55,60)(56,59)(64,65)(66,70)
(67,69)(71,79)(72,78)(73,84)(74,83)(75,82)(76,81)(77,80);;
s2 := ( 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)(43,71)(44,72)(45,73)(46,74)(47,75)(48,76)(49,77)
(50,64)(51,65)(52,66)(53,67)(54,68)(55,69)(56,70)(57,78)(58,79)(59,80)(60,81)
(61,82)(62,83)(63,84);;
s3 := ( 1,43)( 2,44)( 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);;
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*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, 
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)( 8,16)( 9,15)(10,21)(11,20)(12,19)(13,18)
(14,17)(22,23)(24,28)(25,27)(29,37)(30,36)(31,42)(32,41)(33,40)(34,39)(35,38)
(43,44)(45,49)(46,48)(50,58)(51,57)(52,63)(53,62)(54,61)(55,60)(56,59)(64,65)
(66,70)(67,69)(71,79)(72,78)(73,84)(74,83)(75,82)(76,81)(77,80);
s2 := 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)(43,71)(44,72)(45,73)(46,74)(47,75)(48,76)
(49,77)(50,64)(51,65)(52,66)(53,67)(54,68)(55,69)(56,70)(57,78)(58,79)(59,80)
(60,81)(61,82)(62,83)(63,84);
s3 := Sym(84)!( 1,43)( 2,44)( 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);
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*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, 
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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