Polytope of Type {2,2,40,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,2,40,4}*1280a
if this polytope has a name.
Group : SmallGroup(1280,1035864)
Rank : 5
Schlafli Type : {2,2,40,4}
Number of vertices, edges, etc : 2, 2, 40, 80, 4
Order of s₀s₁s₂s₃s₄ : 40
Order of s₀s₁s₂s₃s₄s₃s₂s₁ : 2
Special Properties :
   Degenerate
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {2,2,20,4}*640, {2,2,40,2}*640
   4-fold quotients : {2,2,20,2}*320, {2,2,10,4}*320
   5-fold quotients : {2,2,8,4}*256a
   8-fold quotients : {2,2,10,2}*160
   10-fold quotients : {2,2,4,4}*128, {2,2,8,2}*128
   16-fold quotients : {2,2,5,2}*80
   20-fold quotients : {2,2,2,4}*64, {2,2,4,2}*64
   40-fold quotients : {2,2,2,2}*32
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

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