Polytope of Type {2,28,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,28,6}*672a
if this polytope has a name.
Group : SmallGroup(672,1141)
Rank : 4
Schlafli Type : {2,28,6}
Number of vertices, edges, etc : 2, 28, 84, 6
Order of s₀s₁s₂s₃ : 84
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Degenerate
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {2,28,6,2} of size 1344
Vertex Figure Of :
   {2,2,28,6} of size 1344
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {2,14,6}*336
   3-fold quotients : {2,28,2}*224
   6-fold quotients : {2,14,2}*112
   7-fold quotients : {2,4,6}*96a
   12-fold quotients : {2,7,2}*56
   14-fold quotients : {2,2,6}*48
   21-fold quotients : {2,4,2}*32
   28-fold quotients : {2,2,3}*24
   42-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   2-fold covers : {4,28,6}*1344, {2,56,6}*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)(45,66)(46,72)(47,71)
(48,70)(49,69)(50,68)(51,67)(52,73)(53,79)(54,78)(55,77)(56,76)(57,75)(58,74)
(59,80)(60,86)(61,85)(62,84)(63,83)(64,82)(65,81);;
s2 := ( 3,46)( 4,45)( 5,51)( 6,50)( 7,49)( 8,48)( 9,47)(10,60)(11,59)(12,65)
(13,64)(14,63)(15,62)(16,61)(17,53)(18,52)(19,58)(20,57)(21,56)(22,55)(23,54)
(24,67)(25,66)(26,72)(27,71)(28,70)(29,69)(30,68)(31,81)(32,80)(33,86)(34,85)
(35,84)(36,83)(37,82)(38,74)(39,73)(40,79)(41,78)(42,77)(43,76)(44,75);;
s3 := ( 3,10)( 4,11)( 5,12)( 6,13)( 7,14)( 8,15)( 9,16)(24,31)(25,32)(26,33)
(27,34)(28,35)(29,36)(30,37)(45,52)(46,53)(47,54)(48,55)(49,56)(50,57)(51,58)
(66,73)(67,74)(68,75)(69,76)(70,77)(71,78)(72,79);;
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, 
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*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)(45,66)(46,72)
(47,71)(48,70)(49,69)(50,68)(51,67)(52,73)(53,79)(54,78)(55,77)(56,76)(57,75)
(58,74)(59,80)(60,86)(61,85)(62,84)(63,83)(64,82)(65,81);
s2 := Sym(86)!( 3,46)( 4,45)( 5,51)( 6,50)( 7,49)( 8,48)( 9,47)(10,60)(11,59)
(12,65)(13,64)(14,63)(15,62)(16,61)(17,53)(18,52)(19,58)(20,57)(21,56)(22,55)
(23,54)(24,67)(25,66)(26,72)(27,71)(28,70)(29,69)(30,68)(31,81)(32,80)(33,86)
(34,85)(35,84)(36,83)(37,82)(38,74)(39,73)(40,79)(41,78)(42,77)(43,76)(44,75);
s3 := Sym(86)!( 3,10)( 4,11)( 5,12)( 6,13)( 7,14)( 8,15)( 9,16)(24,31)(25,32)
(26,33)(27,34)(28,35)(29,36)(30,37)(45,52)(46,53)(47,54)(48,55)(49,56)(50,57)
(51,58)(66,73)(67,74)(68,75)(69,76)(70,77)(71,78)(72,79);
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, 
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*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