Polytope of Type {2,82}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,82}*328
if this polytope has a name.
Group : SmallGroup(328,14)
Rank : 3
Schlafli Type : {2,82}
Number of vertices, edges, etc : 2, 82, 82
Order of s₀s₁s₂ : 82
Order of s₀s₁s₂s₁ : 2
Special Properties :
   Degenerate
   Universal
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {2,82,2} of size 656
   {2,82,4} of size 1312
   {2,82,6} of size 1968
Vertex Figure Of :
   {2,2,82} of size 656
   {3,2,82} of size 984
   {4,2,82} of size 1312
   {5,2,82} of size 1640
   {6,2,82} of size 1968
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {2,41}*164
   41-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {2,164}*656, {4,82}*656
   3-fold covers : {6,82}*984, {2,246}*984
   4-fold covers : {4,164}*1312, {8,82}*1312, {2,328}*1312
   5-fold covers : {10,82}*1640, {2,410}*1640
   6-fold covers : {12,82}*1968, {6,164}*1968a, {2,492}*1968, {4,246}*1968a
Permutation Representation (GAP) :

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