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

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

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

Permutation Representation (Magma) :

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

to this polytope