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# Polytope of Type {2,80}

Atlas Canonical Name : {2,80}*320
if this polytope has a name.
Group : SmallGroup(320,529)
Rank : 3
Schlafli Type : {2,80}
Number of vertices, edges, etc : 2, 80, 80
Order of s0s1s2 : 80
Order of s0s1s2s1 : 2
Special Properties :
Degenerate
Universal
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{2,80,2} of size 640
{2,80,4} of size 1280
{2,80,4} of size 1280
{2,80,6} of size 1920
Vertex Figure Of :
{2,2,80} of size 640
{3,2,80} of size 960
{4,2,80} of size 1280
{5,2,80} of size 1600
{6,2,80} of size 1920
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {2,40}*160
4-fold quotients : {2,20}*80
5-fold quotients : {2,16}*64
8-fold quotients : {2,10}*40
10-fold quotients : {2,8}*32
16-fold quotients : {2,5}*20
20-fold quotients : {2,4}*16
40-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
2-fold covers : {4,80}*640a, {2,160}*640
3-fold covers : {6,80}*960, {2,240}*960
4-fold covers : {4,80}*1280a, {8,80}*1280c, {8,80}*1280d, {4,160}*1280a, {4,160}*1280b, {2,320}*1280
5-fold covers : {2,400}*1600, {10,80}*1600a, {10,80}*1600b
6-fold covers : {4,240}*1920a, {12,80}*1920a, {2,480}*1920, {6,160}*1920
Permutation Representation (GAP) :
s0 := (1,2);;
s1 := ( 4, 7)( 5, 6)( 9,12)(10,11)(13,18)(14,22)(15,21)(16,20)(17,19)(23,33)
(24,37)(25,36)(26,35)(27,34)(28,38)(29,42)(30,41)(31,40)(32,39)(43,63)(44,67)
(45,66)(46,65)(47,64)(48,68)(49,72)(50,71)(51,70)(52,69)(53,78)(54,82)(55,81)
(56,80)(57,79)(58,73)(59,77)(60,76)(61,75)(62,74);;
s2 := ( 3,44)( 4,43)( 5,47)( 6,46)( 7,45)( 8,49)( 9,48)(10,52)(11,51)(12,50)
(13,59)(14,58)(15,62)(16,61)(17,60)(18,54)(19,53)(20,57)(21,56)(22,55)(23,74)
(24,73)(25,77)(26,76)(27,75)(28,79)(29,78)(30,82)(31,81)(32,80)(33,64)(34,63)
(35,67)(36,66)(37,65)(38,69)(39,68)(40,72)(41,71)(42,70);;
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 ];;
poly := F / rels;;

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

to this polytope