Polytope of Type {12,14}

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

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

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, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*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*s1*s2*s1*s2 >;

References : None.
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