Polytope of Type {14,12}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {14,12}*336
Also Known As : {14,12|2}. if this polytope has another name.
Group : SmallGroup(336,148)
Rank : 3
Schlafli Type : {14,12}
Number of vertices, edges, etc : 14, 84, 12
Order of s₀s₁s₂ : 84
Order of s₀s₁s₂s₁ : 2
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {14,12,2} of size 672
   {14,12,4} of size 1344
   {14,12,4} of size 1344
   {14,12,4} of size 1344
   {14,12,3} of size 1344
Vertex Figure Of :
   {2,14,12} of size 672
   {4,14,12} of size 1344
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {14,6}*168
   3-fold quotients : {14,4}*112
   6-fold quotients : {14,2}*56
   7-fold quotients : {2,12}*48
   12-fold quotients : {7,2}*28
   14-fold quotients : {2,6}*24
   21-fold quotients : {2,4}*16
   28-fold quotients : {2,3}*12
   42-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {14,24}*672, {28,12}*672
   3-fold covers : {14,36}*1008, {42,12}*1008a, {42,12}*1008b
   4-fold covers : {14,48}*1344, {28,12}*1344a, {28,24}*1344a, {56,12}*1344a, {28,24}*1344b, {56,12}*1344b, {28,12}*1344b
   5-fold covers : {14,60}*1680, {70,12}*1680
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)(43,65)
(44,64)(45,70)(46,69)(47,68)(48,67)(49,66)(50,79)(51,78)(52,84)(53,83)(54,82)
(55,81)(56,80)(57,72)(58,71)(59,77)(60,76)(61,75)(62,74)(63,73);;
s2 := ( 1,50)( 2,51)( 3,52)( 4,53)( 5,54)( 6,55)( 7,56)( 8,43)( 9,44)(10,45)
(11,46)(12,47)(13,48)(14,49)(15,57)(16,58)(17,59)(18,60)(19,61)(20,62)(21,63)
(22,71)(23,72)(24,73)(25,74)(26,75)(27,76)(28,77)(29,64)(30,65)(31,66)(32,67)
(33,68)(34,69)(35,70)(36,78)(37,79)(38,80)(39,81)(40,82)(41,83)(42,84);;
poly := Group([s0,s1,s2]);;

Finitely Presented Group Representation (GAP) :

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

Finitely Presented Group Representation (Magma) :

poly<s0,s1,s2> := Group< s0,s1,s2 | s0*s0, s1*s1, s2*s2, 
s0*s2*s0*s2, s0*s1*s2*s1*s0*s1*s2*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*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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