Polytope of Type {3,6,10}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,6,10}*1920
Also Known As : {{3,6}₈,{6,10|2}}. if this polytope has another name.
Group : SmallGroup(1920,238599)
Rank : 4
Schlafli Type : {3,6,10}
Number of vertices, edges, etc : 16, 48, 160, 10
Order of s₀s₁s₂s₃ : 40
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   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) :
   4-fold quotients : {3,6,10}*480
   5-fold quotients : {3,6,2}*384
   20-fold quotients : {3,6,2}*96
   40-fold quotients : {3,3,2}*48
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope