Polytope of Type {2,6,28}

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