Polytope of Type {2,14,12}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,14,12}*672
if this polytope has a name.
Group : SmallGroup(672,1140)
Rank : 4
Schlafli Type : {2,14,12}
Number of vertices, edges, etc : 2, 14, 84, 12
Order of s₀s₁s₂s₃ : 84
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Degenerate
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {2,14,12,2} of size 1344
Vertex Figure Of :
   {2,2,14,12} of size 1344
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {2,14,6}*336
   3-fold quotients : {2,14,4}*224
   6-fold quotients : {2,14,2}*112
   7-fold quotients : {2,2,12}*96
   12-fold quotients : {2,7,2}*56
   14-fold quotients : {2,2,6}*48
   21-fold quotients : {2,2,4}*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, {2,14,24}*1344, {2,28,12}*1344
Permutation Representation (GAP) :

s0 := (1,2);;
s1 := ( 4, 9)( 5, 8)( 6, 7)(11,16)(12,15)(13,14)(18,23)(19,22)(20,21)(25,30)
(26,29)(27,28)(32,37)(33,36)(34,35)(39,44)(40,43)(41,42)(46,51)(47,50)(48,49)
(53,58)(54,57)(55,56)(60,65)(61,64)(62,63)(67,72)(68,71)(69,70)(74,79)(75,78)
(76,77)(81,86)(82,85)(83,84);;
s2 := ( 3, 4)( 5, 9)( 6, 8)(10,18)(11,17)(12,23)(13,22)(14,21)(15,20)(16,19)
(24,25)(26,30)(27,29)(31,39)(32,38)(33,44)(34,43)(35,42)(36,41)(37,40)(45,67)
(46,66)(47,72)(48,71)(49,70)(50,69)(51,68)(52,81)(53,80)(54,86)(55,85)(56,84)
(57,83)(58,82)(59,74)(60,73)(61,79)(62,78)(63,77)(64,76)(65,75);;
s3 := ( 3,52)( 4,53)( 5,54)( 6,55)( 7,56)( 8,57)( 9,58)(10,45)(11,46)(12,47)
(13,48)(14,49)(15,50)(16,51)(17,59)(18,60)(19,61)(20,62)(21,63)(22,64)(23,65)
(24,73)(25,74)(26,75)(27,76)(28,77)(29,78)(30,79)(31,66)(32,67)(33,68)(34,69)
(35,70)(36,71)(37,72)(38,80)(39,81)(40,82)(41,83)(42,84)(43,85)(44,86);;
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*s1*s0*s1, 
s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s1*s2*s3*s2*s1*s2*s3*s2, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*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(86)!(1,2);
s1 := Sym(86)!( 4, 9)( 5, 8)( 6, 7)(11,16)(12,15)(13,14)(18,23)(19,22)(20,21)
(25,30)(26,29)(27,28)(32,37)(33,36)(34,35)(39,44)(40,43)(41,42)(46,51)(47,50)
(48,49)(53,58)(54,57)(55,56)(60,65)(61,64)(62,63)(67,72)(68,71)(69,70)(74,79)
(75,78)(76,77)(81,86)(82,85)(83,84);
s2 := Sym(86)!( 3, 4)( 5, 9)( 6, 8)(10,18)(11,17)(12,23)(13,22)(14,21)(15,20)
(16,19)(24,25)(26,30)(27,29)(31,39)(32,38)(33,44)(34,43)(35,42)(36,41)(37,40)
(45,67)(46,66)(47,72)(48,71)(49,70)(50,69)(51,68)(52,81)(53,80)(54,86)(55,85)
(56,84)(57,83)(58,82)(59,74)(60,73)(61,79)(62,78)(63,77)(64,76)(65,75);
s3 := Sym(86)!( 3,52)( 4,53)( 5,54)( 6,55)( 7,56)( 8,57)( 9,58)(10,45)(11,46)
(12,47)(13,48)(14,49)(15,50)(16,51)(17,59)(18,60)(19,61)(20,62)(21,63)(22,64)
(23,65)(24,73)(25,74)(26,75)(27,76)(28,77)(29,78)(30,79)(31,66)(32,67)(33,68)
(34,69)(35,70)(36,71)(37,72)(38,80)(39,81)(40,82)(41,83)(42,84)(43,85)(44,86);
poly := sub<Sym(86)|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*s1*s0*s1, s0*s2*s0*s2, s0*s3*s0*s3, 
s1*s3*s1*s3, s1*s2*s3*s2*s1*s2*s3*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*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 >;

to this polytope