Polytope of Type {8,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {8,6}*1296
if this polytope has a name.
Group : SmallGroup(1296,3509)
Rank : 3
Schlafli Type : {8,6}
Number of vertices, edges, etc : 108, 324, 81
Order of s₀s₁s₂ : 8
Order of s₀s₁s₂s₁ : 6
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Non-Orientable
   Self-Petrie
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) :
   No Regular Quotients.
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope