Polytope of Type {2,40,4}

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

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

to this polytope