Polytope of Type {5,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {5,6}*1620
if this polytope has a name.
Group : SmallGroup(1620,422)
Rank : 3
Schlafli Type : {5,6}
Number of vertices, edges, etc : 135, 405, 162
Order of s₀s₁s₂ : 10
Order of s₀s₁s₂s₁ : 6
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   81-fold quotients : {5,2}*20
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope