Polytope of Type {4,20,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,20,6}*1920b
if this polytope has a name.
Group : SmallGroup(1920,238598)
Rank : 4
Schlafli Type : {4,20,6}
Number of vertices, edges, etc : 8, 80, 120, 6
Order of s₀s₁s₂s₃ : 30
Order of s₀s₁s₂s₃s₂s₁ : 4
Special Properties :
   Non-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) :
   4-fold quotients : {2,20,6}*480b
   5-fold quotients : {4,4,6}*384c
   10-fold quotients : {4,4,3}*192a
   20-fold quotients : {2,4,6}*96b
   40-fold quotients : {2,4,3}*48
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
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