Polytope of Type {4,40}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,40}*640b
if this polytope has a name.
Group : SmallGroup(640,2005)
Rank : 3
Schlafli Type : {4,40}
Number of vertices, edges, etc : 8, 160, 80
Order of s₀s₁s₂ : 20
Order of s₀s₁s₂s₁ : 4
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {4,40,2} of size 1280
Vertex Figure Of :
   {2,4,40} of size 1280
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {4,20}*320
   4-fold quotients : {4,20}*160
   5-fold quotients : {4,8}*128b
   8-fold quotients : {2,20}*80, {4,10}*80
   10-fold quotients : {4,4}*64
   16-fold quotients : {2,10}*40
   20-fold quotients : {4,4}*32
   32-fold quotients : {2,5}*20
   40-fold quotients : {2,4}*16, {4,2}*16
   80-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {8,40}*1280b, {4,40}*1280a, {8,40}*1280c, {4,40}*1280b, {8,40}*1280e, {4,40}*1280d, {8,40}*1280g
   3-fold covers : {4,120}*1920b, {12,40}*1920b
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope