Polytope of Type {4,14,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,14,6}*672
Also Known As : {{4,14|2},{14,6|2}}. if this polytope has another name.
Group : SmallGroup(672,1150)
Rank : 4
Schlafli Type : {4,14,6}
Number of vertices, edges, etc : 4, 28, 42, 6
Order of s₀s₁s₂s₃ : 84
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {4,14,6,2} of size 1344
Vertex Figure Of :
   {2,4,14,6} of size 1344
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {2,14,6}*336
   3-fold quotients : {4,14,2}*224
   6-fold quotients : {2,14,2}*112
   7-fold quotients : {4,2,6}*96
   12-fold quotients : {2,7,2}*56
   14-fold quotients : {4,2,3}*48, {2,2,6}*48
   21-fold quotients : {4,2,2}*32
   28-fold quotients : {2,2,3}*24
   42-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   2-fold covers : {4,14,12}*1344, {4,28,6}*1344, {8,14,6}*1344
Permutation Representation (GAP) :

s0 := (43,64)(44,65)(45,66)(46,67)(47,68)(48,69)(49,70)(50,71)(51,72)(52,73)
(53,74)(54,75)(55,76)(56,77)(57,78)(58,79)(59,80)(60,81)(61,82)(62,83)
(63,84);;
s1 := ( 1,43)( 2,49)( 3,48)( 4,47)( 5,46)( 6,45)( 7,44)( 8,50)( 9,56)(10,55)
(11,54)(12,53)(13,52)(14,51)(15,57)(16,63)(17,62)(18,61)(19,60)(20,59)(21,58)
(22,64)(23,70)(24,69)(25,68)(26,67)(27,66)(28,65)(29,71)(30,77)(31,76)(32,75)
(33,74)(34,73)(35,72)(36,78)(37,84)(38,83)(39,82)(40,81)(41,80)(42,79);;
s2 := ( 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);;
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*s0*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s0*s1*s2*s1, s1*s2*s3*s2*s1*s2*s3*s2, 
s2*s3*s2*s3*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 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(84)!(43,64)(44,65)(45,66)(46,67)(47,68)(48,69)(49,70)(50,71)(51,72)
(52,73)(53,74)(54,75)(55,76)(56,77)(57,78)(58,79)(59,80)(60,81)(61,82)(62,83)
(63,84);
s1 := Sym(84)!( 1,43)( 2,49)( 3,48)( 4,47)( 5,46)( 6,45)( 7,44)( 8,50)( 9,56)
(10,55)(11,54)(12,53)(13,52)(14,51)(15,57)(16,63)(17,62)(18,61)(19,60)(20,59)
(21,58)(22,64)(23,70)(24,69)(25,68)(26,67)(27,66)(28,65)(29,71)(30,77)(31,76)
(32,75)(33,74)(34,73)(35,72)(36,78)(37,84)(38,83)(39,82)(40,81)(41,80)(42,79);
s2 := 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);
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*s0*s1*s0*s1*s0*s1, s0*s1*s2*s1*s0*s1*s2*s1, 
s1*s2*s3*s2*s1*s2*s3*s2, s2*s3*s2*s3*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 >;

References : None.
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