Polytope of Type {6,4,14}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,4,14}*672
Also Known As : {{6,4|2},{4,14|2}}. if this polytope has another name.
Group : SmallGroup(672,1150)
Rank : 4
Schlafli Type : {6,4,14}
Number of vertices, edges, etc : 6, 12, 28, 14
Order of s₀s₁s₂s₃ : 84
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {6,4,14,2} of size 1344
Vertex Figure Of :
   {2,6,4,14} of size 1344
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {6,2,14}*336
   3-fold quotients : {2,4,14}*224
   4-fold quotients : {3,2,14}*168, {6,2,7}*168
   6-fold quotients : {2,2,14}*112
   7-fold quotients : {6,4,2}*96a
   8-fold quotients : {3,2,7}*84
   12-fold quotients : {2,2,7}*56
   14-fold quotients : {6,2,2}*48
   21-fold quotients : {2,4,2}*32
   28-fold quotients : {3,2,2}*24
   42-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   2-fold covers : {12,4,14}*1344, {6,4,28}*1344, {6,8,14}*1344
Permutation Representation (GAP) :

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

References : None.
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