Polytope of Type {6,8}

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

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

Permutation Representation (Magma) :

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

References : None.
to this polytope