Polytope of Type {2,2,4,20}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,2,4,20}*640
if this polytope has a name.
Group : SmallGroup(640,19674)
Rank : 5
Schlafli Type : {2,2,4,20}
Number of vertices, edges, etc : 2, 2, 4, 40, 20
Order of s₀s₁s₂s₃s₄ : 20
Order of s₀s₁s₂s₃s₄s₃s₂s₁ : 2
Special Properties :
   Degenerate
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {2,2,4,20,2} of size 1280
Vertex Figure Of :
   {2,2,2,4,20} of size 1280
   {3,2,2,4,20} of size 1920
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {2,2,2,20}*320, {2,2,4,10}*320
   4-fold quotients : {2,2,2,10}*160
   5-fold quotients : {2,2,4,4}*128
   8-fold quotients : {2,2,2,5}*80
   10-fold quotients : {2,2,2,4}*64, {2,2,4,2}*64
   20-fold quotients : {2,2,2,2}*32
Covers (Minimal Covers in Boldface) :
   2-fold covers : {2,4,4,20}*1280, {4,2,4,20}*1280, {2,2,8,20}*1280a, {2,2,4,40}*1280a, {2,2,8,20}*1280b, {2,2,4,40}*1280b, {2,2,4,20}*1280
   3-fold covers : {2,2,4,60}*1920a, {2,6,4,20}*1920, {6,2,4,20}*1920, {2,2,12,20}*1920
Permutation Representation (GAP) :

s0 := (1,2);;
s1 := (3,4);;
s2 := ( 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)(21,61)(22,62)(23,63)(24,64)(25,70)
(26,71)(27,72)(28,73)(29,74)(30,65)(31,66)(32,67)(33,68)(34,69)(35,80)(36,81)
(37,82)(38,83)(39,84)(40,75)(41,76)(42,77)(43,78)(44,79);;
s3 := ( 5,25)( 6,29)( 7,28)( 8,27)( 9,26)(10,30)(11,34)(12,33)(13,32)(14,31)
(15,35)(16,39)(17,38)(18,37)(19,36)(20,40)(21,44)(22,43)(23,42)(24,41)(45,65)
(46,69)(47,68)(48,67)(49,66)(50,70)(51,74)(52,73)(53,72)(54,71)(55,75)(56,79)
(57,78)(58,77)(59,76)(60,80)(61,84)(62,83)(63,82)(64,81);;
s4 := ( 5, 6)( 7, 9)(10,11)(12,14)(15,16)(17,19)(20,21)(22,24)(25,36)(26,35)
(27,39)(28,38)(29,37)(30,41)(31,40)(32,44)(33,43)(34,42)(45,46)(47,49)(50,51)
(52,54)(55,56)(57,59)(60,61)(62,64)(65,76)(66,75)(67,79)(68,78)(69,77)(70,81)
(71,80)(72,84)(73,83)(74,82);;
poly := Group([s0,s1,s2,s3,s4]);;

Finitely Presented Group Representation (GAP) :

F := FreeGroup("s0","s1","s2","s3","s4");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  s4 := F.5;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s4*s4, s0*s1*s0*s1, 
s0*s2*s0*s2, s1*s2*s1*s2, s0*s3*s0*s3, 
s1*s3*s1*s3, s0*s4*s0*s4, s1*s4*s1*s4, 
s2*s4*s2*s4, s2*s3*s2*s3*s2*s3*s2*s3, 
s2*s3*s4*s3*s2*s3*s4*s3, s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(84)!(1,2);
s1 := Sym(84)!(3,4);
s2 := Sym(84)!( 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)(21,61)(22,62)(23,63)(24,64)
(25,70)(26,71)(27,72)(28,73)(29,74)(30,65)(31,66)(32,67)(33,68)(34,69)(35,80)
(36,81)(37,82)(38,83)(39,84)(40,75)(41,76)(42,77)(43,78)(44,79);
s3 := Sym(84)!( 5,25)( 6,29)( 7,28)( 8,27)( 9,26)(10,30)(11,34)(12,33)(13,32)
(14,31)(15,35)(16,39)(17,38)(18,37)(19,36)(20,40)(21,44)(22,43)(23,42)(24,41)
(45,65)(46,69)(47,68)(48,67)(49,66)(50,70)(51,74)(52,73)(53,72)(54,71)(55,75)
(56,79)(57,78)(58,77)(59,76)(60,80)(61,84)(62,83)(63,82)(64,81);
s4 := Sym(84)!( 5, 6)( 7, 9)(10,11)(12,14)(15,16)(17,19)(20,21)(22,24)(25,36)
(26,35)(27,39)(28,38)(29,37)(30,41)(31,40)(32,44)(33,43)(34,42)(45,46)(47,49)
(50,51)(52,54)(55,56)(57,59)(60,61)(62,64)(65,76)(66,75)(67,79)(68,78)(69,77)
(70,81)(71,80)(72,84)(73,83)(74,82);
poly := sub<Sym(84)|s0,s1,s2,s3,s4>;

Finitely Presented Group Representation (Magma) :

poly<s0,s1,s2,s3,s4> := Group< s0,s1,s2,s3,s4 | s0*s0, s1*s1, s2*s2, 
s3*s3, s4*s4, s0*s1*s0*s1, s0*s2*s0*s2, 
s1*s2*s1*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s0*s4*s0*s4, s1*s4*s1*s4, s2*s4*s2*s4, 
s2*s3*s2*s3*s2*s3*s2*s3, s2*s3*s4*s3*s2*s3*s4*s3, 
s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4 >;

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