Polytope of Type {8,20}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {8,20}*320a
Also Known As : {8,20|2}. if this polytope has another name.
Group : SmallGroup(320,400)
Rank : 3
Schlafli Type : {8,20}
Number of vertices, edges, etc : 8, 80, 20
Order of s₀s₁s₂ : 40
Order of s₀s₁s₂s₁ : 2
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {8,20,2} of size 640
   {8,20,4} of size 1280
   {8,20,6} of size 1920
Vertex Figure Of :
   {2,8,20} of size 640
   {4,8,20} of size 1280
   {4,8,20} of size 1280
   {6,8,20} of size 1920
   {3,8,20} of size 1920
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {4,20}*160, {8,10}*160
   4-fold quotients : {2,20}*80, {4,10}*80
   5-fold quotients : {8,4}*64a
   8-fold quotients : {2,10}*40
   10-fold quotients : {4,4}*32, {8,2}*32
   16-fold quotients : {2,5}*20
   20-fold quotients : {2,4}*16, {4,2}*16
   40-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {8,40}*640b, {8,20}*640a, {8,40}*640d, {16,20}*640a, {16,20}*640b
   3-fold covers : {24,20}*960a, {8,60}*960a
   4-fold covers : {8,40}*1280a, {8,20}*1280a, {8,40}*1280c, {16,20}*1280a, {16,20}*1280b, {8,80}*1280a, {8,80}*1280b, {16,40}*1280c, {8,80}*1280d, {16,40}*1280d, {16,40}*1280e, {8,80}*1280f, {16,40}*1280f, {32,20}*1280a, {32,20}*1280b
   5-fold covers : {8,100}*1600a, {40,20}*1600b, {40,20}*1600c
   6-fold covers : {8,60}*1920a, {24,20}*1920a, {8,120}*1920a, {8,120}*1920c, {24,40}*1920a, {24,40}*1920b, {16,60}*1920a, {48,20}*1920a, {16,60}*1920b, {48,20}*1920b
Permutation Representation (GAP) :

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

References : None.
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