Polytope of Type {2,12,14}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,12,14}*672
if this polytope has a name.
Group : SmallGroup(672,1140)
Rank : 4
Schlafli Type : {2,12,14}
Number of vertices, edges, etc : 2, 12, 84, 14
Order of s₀s₁s₂s₃ : 84
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Degenerate
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {2,12,14,2} of size 1344
Vertex Figure Of :
   {2,2,12,14} of size 1344
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {2,6,14}*336
   3-fold quotients : {2,4,14}*224
   6-fold quotients : {2,2,14}*112
   7-fold quotients : {2,12,2}*96
   12-fold quotients : {2,2,7}*56
   14-fold quotients : {2,6,2}*48
   21-fold quotients : {2,4,2}*32
   28-fold quotients : {2,3,2}*24
   42-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   2-fold covers : {4,12,14}*1344a, {2,24,14}*1344, {2,12,28}*1344
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

to this polytope