Polytope of Type {82}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {82}*164
Also Known As : 82-gon, {82}. if this polytope has another name.
Group : SmallGroup(164,4)
Rank : 2
Schlafli Type : {82}
Number of vertices, edges, etc : 82, 82
Order of s₀s₁ : 82
Special Properties :
   Universal
   Spherical
   Locally Spherical
   Orientable
   Self-Dual
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {82,2} of size 328
   {82,4} of size 656
   {82,6} of size 984
   {82,8} of size 1312
   {82,10} of size 1640
   {82,12} of size 1968
Vertex Figure Of :
   {2,82} of size 328
   {4,82} of size 656
   {6,82} of size 984
   {8,82} of size 1312
   {10,82} of size 1640
   {12,82} of size 1968
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {41}*82
   41-fold quotients : {2}*4
Covers (Minimal Covers in Boldface) :
   2-fold covers : {164}*328
   3-fold covers : {246}*492
   4-fold covers : {328}*656
   5-fold covers : {410}*820
   6-fold covers : {492}*984
   7-fold covers : {574}*1148
   8-fold covers : {656}*1312
   9-fold covers : {738}*1476
   10-fold covers : {820}*1640
   11-fold covers : {902}*1804
   12-fold covers : {984}*1968
Permutation Representation (GAP) :

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

Finitely Presented Group Representation (GAP) :

F := FreeGroup("s0","s1");;
s0 := F.1;;  s1 := F.2;;  
rels := [ s0*s0, s1*s1, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;

Permutation Representation (Magma) :

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

Finitely Presented Group Representation (Magma) :

poly<s0,s1> := Group< s0,s1 | s0*s0, s1*s1, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >;

References : None.
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