Polytope of Type {2,2,18,9}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,2,18,9}*1296
if this polytope has a name.
Group : SmallGroup(1296,1857)
Rank : 5
Schlafli Type : {2,2,18,9}
Number of vertices, edges, etc : 2, 2, 18, 81, 9
Order of s₀s₁s₂s₃s₄ : 18
Order of s₀s₁s₂s₃s₄s₃s₂s₁ : 2
Special Properties :
   Degenerate
   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) :
   3-fold quotients : {2,2,6,9}*432
   9-fold quotients : {2,2,2,9}*144, {2,2,6,3}*144
   27-fold quotients : {2,2,2,3}*48
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := (1,2);;
s1 := (3,4);;
s2 := ( 8,11)( 9,12)(10,13)(17,20)(18,21)(19,22)(26,29)(27,30)(28,31)(32,62)
(33,63)(34,64)(35,59)(36,60)(37,61)(38,65)(39,66)(40,67)(41,71)(42,72)(43,73)
(44,68)(45,69)(46,70)(47,74)(48,75)(49,76)(50,80)(51,81)(52,82)(53,77)(54,78)
(55,79)(56,83)(57,84)(58,85);;
s3 := ( 5,32)( 6,34)( 7,33)( 8,38)( 9,40)(10,39)(11,35)(12,37)(13,36)(14,51)
(15,50)(16,52)(17,57)(18,56)(19,58)(20,54)(21,53)(22,55)(23,42)(24,41)(25,43)
(26,48)(27,47)(28,49)(29,45)(30,44)(31,46)(59,62)(60,64)(61,63)(66,67)(68,81)
(69,80)(70,82)(71,78)(72,77)(73,79)(74,84)(75,83)(76,85);;
s4 := ( 5,14)( 6,16)( 7,15)( 8,20)( 9,22)(10,21)(11,17)(12,19)(13,18)(23,24)
(26,30)(27,29)(28,31)(32,71)(33,73)(34,72)(35,68)(36,70)(37,69)(38,74)(39,76)
(40,75)(41,62)(42,64)(43,63)(44,59)(45,61)(46,60)(47,65)(48,67)(49,66)(50,81)
(51,80)(52,82)(53,78)(54,77)(55,79)(56,84)(57,83)(58,85);;
poly := Group([s0,s1,s2,s3,s4]);;

Finitely Presented Group Representation (GAP) :

F := FreeGroup("s0","s1","s2","s3","s4");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  s4 := F.5;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s4*s4, s0*s1*s0*s1, 
s0*s2*s0*s2, s1*s2*s1*s2, s0*s3*s0*s3, 
s1*s3*s1*s3, s0*s4*s0*s4, s1*s4*s1*s4, 
s2*s4*s2*s4, s4*s2*s3*s2*s3*s4*s2*s3*s2*s3, 
s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(85)!(1,2);
s1 := Sym(85)!(3,4);
s2 := Sym(85)!( 8,11)( 9,12)(10,13)(17,20)(18,21)(19,22)(26,29)(27,30)(28,31)
(32,62)(33,63)(34,64)(35,59)(36,60)(37,61)(38,65)(39,66)(40,67)(41,71)(42,72)
(43,73)(44,68)(45,69)(46,70)(47,74)(48,75)(49,76)(50,80)(51,81)(52,82)(53,77)
(54,78)(55,79)(56,83)(57,84)(58,85);
s3 := Sym(85)!( 5,32)( 6,34)( 7,33)( 8,38)( 9,40)(10,39)(11,35)(12,37)(13,36)
(14,51)(15,50)(16,52)(17,57)(18,56)(19,58)(20,54)(21,53)(22,55)(23,42)(24,41)
(25,43)(26,48)(27,47)(28,49)(29,45)(30,44)(31,46)(59,62)(60,64)(61,63)(66,67)
(68,81)(69,80)(70,82)(71,78)(72,77)(73,79)(74,84)(75,83)(76,85);
s4 := Sym(85)!( 5,14)( 6,16)( 7,15)( 8,20)( 9,22)(10,21)(11,17)(12,19)(13,18)
(23,24)(26,30)(27,29)(28,31)(32,71)(33,73)(34,72)(35,68)(36,70)(37,69)(38,74)
(39,76)(40,75)(41,62)(42,64)(43,63)(44,59)(45,61)(46,60)(47,65)(48,67)(49,66)
(50,81)(51,80)(52,82)(53,78)(54,77)(55,79)(56,84)(57,83)(58,85);
poly := sub<Sym(85)|s0,s1,s2,s3,s4>;

Finitely Presented Group Representation (Magma) :

poly<s0,s1,s2,s3,s4> := Group< s0,s1,s2,s3,s4 | s0*s0, s1*s1, s2*s2, 
s3*s3, s4*s4, s0*s1*s0*s1, s0*s2*s0*s2, 
s1*s2*s1*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s0*s4*s0*s4, s1*s4*s1*s4, s2*s4*s2*s4, 
s4*s2*s3*s2*s3*s4*s2*s3*s2*s3, s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4 >;

to this polytope