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

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

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

Permutation Representation (Magma) :

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

to this polytope