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# Polytope of Type {82,2}

Atlas Canonical Name : {82,2}*328
if this polytope has a name.
Group : SmallGroup(328,14)
Rank : 3
Schlafli Type : {82,2}
Number of vertices, edges, etc : 82, 82, 2
Order of s0s1s2 : 82
Order of s0s1s2s1 : 2
Special Properties :
Degenerate
Universal
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Flat
Self-Petrie
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Facet Of :
{82,2,2} of size 656
{82,2,3} of size 984
{82,2,4} of size 1312
{82,2,5} of size 1640
{82,2,6} of size 1968
Vertex Figure Of :
{2,82,2} of size 656
{4,82,2} of size 1312
{6,82,2} of size 1968
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {41,2}*164
41-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
2-fold covers : {164,2}*656, {82,4}*656
3-fold covers : {82,6}*984, {246,2}*984
4-fold covers : {164,4}*1312, {82,8}*1312, {328,2}*1312
5-fold covers : {82,10}*1640, {410,2}*1640
6-fold covers : {82,12}*1968, {164,6}*1968a, {492,2}*1968, {246,4}*1968a
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);;
s2 := (83,84);;
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*s2*s0*s2, s1*s2*s1*s2,
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(84)!( 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(84)!( 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);
s2 := Sym(84)!(83,84);
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*s2*s0*s2, s1*s2*s1*s2, 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 >;

```

to this polytope