Polytope of Type {20,4}

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

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

Permutation Representation (Magma) :

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

References : None.
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