Polytope of Type {4,20}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,20}*160
Also Known As : {4,20|2}. if this polytope has another name.
Group : SmallGroup(160,103)
Rank : 3
Schlafli Type : {4,20}
Number of vertices, edges, etc : 4, 40, 20
Order of s₀s₁s₂ : 20
Order of s₀s₁s₂s₁ : 2
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {4,20,2} of size 320
   {4,20,4} of size 640
   {4,20,6} of size 960
   {4,20,8} of size 1280
   {4,20,8} of size 1280
   {4,20,4} of size 1280
   {4,20,6} of size 1440
   {4,20,10} of size 1600
   {4,20,10} of size 1600
   {4,20,10} of size 1600
   {4,20,12} of size 1920
   {4,20,6} of size 1920
   {4,20,6} of size 1920
   {4,20,6} of size 1920
   {4,20,10} of size 1920
   {4,20,10} of size 1920
   {4,20,3} of size 1920
   {4,20,5} of size 1920
Vertex Figure Of :
   {2,4,20} of size 320
   {4,4,20} of size 640
   {6,4,20} of size 960
   {3,4,20} of size 960
   {8,4,20} of size 1280
   {8,4,20} of size 1280
   {4,4,20} of size 1280
   {6,4,20} of size 1440
   {10,4,20} of size 1600
   {12,4,20} of size 1920
   {6,4,20} of size 1920
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {2,20}*80, {4,10}*80
   4-fold quotients : {2,10}*40
   5-fold quotients : {4,4}*32
   8-fold quotients : {2,5}*20
   10-fold quotients : {2,4}*16, {4,2}*16
   20-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {4,40}*320a, {4,20}*320, {4,40}*320b, {8,20}*320a, {8,20}*320b
   3-fold covers : {12,20}*480, {4,60}*480a
   4-fold covers : {4,40}*640a, {8,40}*640a, {8,40}*640b, {8,20}*640a, {8,40}*640c, {8,40}*640d, {4,80}*640a, {4,80}*640b, {4,20}*640a, {4,40}*640b, {8,20}*640b, {16,20}*640a, {16,20}*640b
   5-fold covers : {4,100}*800, {20,20}*800a, {20,20}*800b
   6-fold covers : {12,20}*960a, {24,20}*960a, {12,40}*960a, {24,20}*960b, {12,40}*960b, {4,120}*960a, {4,60}*960a, {4,120}*960b, {8,60}*960a, {8,60}*960b
   7-fold covers : {28,20}*1120, {4,140}*1120
   8-fold covers : {8,40}*1280a, {8,20}*1280a, {8,40}*1280b, {4,40}*1280a, {8,40}*1280c, {8,40}*1280d, {16,20}*1280a, {4,80}*1280a, {16,20}*1280b, {4,80}*1280b, {8,80}*1280a, {16,40}*1280a, {8,80}*1280b, {16,40}*1280b, {16,40}*1280c, {8,80}*1280c, {8,80}*1280d, {16,40}*1280d, {16,40}*1280e, {8,80}*1280e, {8,80}*1280f, {16,40}*1280f, {32,20}*1280a, {4,160}*1280a, {32,20}*1280b, {4,160}*1280b, {4,20}*1280a, {4,40}*1280b, {8,20}*1280b, {8,20}*1280c, {8,40}*1280e, {4,40}*1280c, {4,40}*1280d, {8,20}*1280d, {8,40}*1280f, {8,40}*1280g, {8,40}*1280h
   9-fold covers : {36,20}*1440, {4,180}*1440a, {12,60}*1440a, {12,60}*1440b, {12,60}*1440c, {4,20}*1440, {4,60}*1440, {12,20}*1440
   10-fold covers : {4,200}*1600a, {4,100}*1600, {4,200}*1600b, {8,100}*1600a, {8,100}*1600b, {40,20}*1600a, {20,20}*1600a, {20,20}*1600b, {40,20}*1600b, {20,40}*1600c, {20,40}*1600d, {40,20}*1600c, {20,40}*1600e, {20,40}*1600f, {40,20}*1600e
   11-fold covers : {44,20}*1760, {4,220}*1760
   12-fold covers : {8,60}*1920a, {4,120}*1920a, {12,40}*1920a, {24,20}*1920a, {8,120}*1920a, {8,120}*1920b, {8,120}*1920c, {24,40}*1920a, {24,40}*1920b, {24,40}*1920c, {8,120}*1920d, {24,40}*1920d, {16,60}*1920a, {4,240}*1920a, {12,80}*1920a, {48,20}*1920a, {16,60}*1920b, {4,240}*1920b, {12,80}*1920b, {48,20}*1920b, {4,60}*1920a, {4,120}*1920b, {8,60}*1920b, {12,40}*1920b, {24,20}*1920b, {12,20}*1920a, {12,20}*1920c, {12,60}*1920c, {4,60}*1920d
Permutation Representation (GAP) :

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