Polytope of Type {2,4,40}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,4,40}*640a
if this polytope has a name.
Group : SmallGroup(640,12419)
Rank : 4
Schlafli Type : {2,4,40}
Number of vertices, edges, etc : 2, 4, 80, 40
Order of s₀s₁s₂s₃ : 40
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Degenerate
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {2,4,40,2} of size 1280
Vertex Figure Of :
   {2,2,4,40} of size 1280
   {3,2,4,40} of size 1920
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {2,4,20}*320, {2,2,40}*320
   4-fold quotients : {2,2,20}*160, {2,4,10}*160
   5-fold quotients : {2,4,8}*128a
   8-fold quotients : {2,2,10}*80
   10-fold quotients : {2,4,4}*64, {2,2,8}*64
   16-fold quotients : {2,2,5}*40
   20-fold quotients : {2,2,4}*32, {2,4,2}*32
   40-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   2-fold covers : {2,4,40}*1280a, {2,8,40}*1280b, {2,8,40}*1280c, {4,4,40}*1280a, {2,4,80}*1280a, {2,4,80}*1280b
   3-fold covers : {2,4,120}*1920a, {6,4,40}*1920a, {2,12,40}*1920a
Permutation Representation (GAP) :

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

Finitely Presented Group Representation (GAP) :

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

Finitely Presented Group Representation (Magma) :

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

to this polytope