Polytope of Type {2,2,18,3}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,2,18,3}*1296
if this polytope has a name.
Group : SmallGroup(1296,1862)
Rank : 5
Schlafli Type : {2,2,18,3}
Number of vertices, edges, etc : 2, 2, 54, 81, 9
Order of s₀s₁s₂s₃s₄ : 6
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 : {2,2,6,3}*432
   9-fold quotients : {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 := ( 6, 7)( 8,11)( 9,13)(10,12)(15,16)(17,20)(18,22)(19,21)(24,25)(26,29)
(27,31)(28,30)(32,61)(33,60)(34,59)(35,67)(36,66)(37,65)(38,64)(39,63)(40,62)
(41,70)(42,69)(43,68)(44,76)(45,75)(46,74)(47,73)(48,72)(49,71)(50,79)(51,78)
(52,77)(53,85)(54,84)(55,83)(56,82)(57,81)(58,80);;
s3 := ( 5,32)( 6,34)( 7,33)( 8,37)( 9,36)(10,35)(11,39)(12,38)(13,40)(14,57)
(15,56)(16,58)(17,50)(18,52)(19,51)(20,55)(21,54)(22,53)(23,44)(24,46)(25,45)
(26,49)(27,48)(28,47)(29,42)(30,41)(31,43)(59,61)(62,63)(66,67)(68,83)(69,85)
(70,84)(71,79)(72,78)(73,77)(74,81)(75,80)(76,82);;
s4 := ( 5,14)( 6,16)( 7,15)( 8,17)( 9,19)(10,18)(11,20)(12,22)(13,21)(24,25)
(27,28)(30,31)(32,70)(33,69)(34,68)(35,73)(36,72)(37,71)(38,76)(39,75)(40,74)
(41,61)(42,60)(43,59)(44,64)(45,63)(46,62)(47,67)(48,66)(49,65)(50,79)(51,78)
(52,77)(53,82)(54,81)(55,80)(56,85)(57,84)(58,83);;
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, s3*s4*s3*s4*s3*s4, s2*s4*s3*s2*s4*s3*s2*s4*s3*s2*s4*s3*s2*s4*s3*s2*s4*s3, 
s4*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s4*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)(15,16)(17,20)(18,22)(19,21)(24,25)
(26,29)(27,31)(28,30)(32,61)(33,60)(34,59)(35,67)(36,66)(37,65)(38,64)(39,63)
(40,62)(41,70)(42,69)(43,68)(44,76)(45,75)(46,74)(47,73)(48,72)(49,71)(50,79)
(51,78)(52,77)(53,85)(54,84)(55,83)(56,82)(57,81)(58,80);
s3 := Sym(85)!( 5,32)( 6,34)( 7,33)( 8,37)( 9,36)(10,35)(11,39)(12,38)(13,40)
(14,57)(15,56)(16,58)(17,50)(18,52)(19,51)(20,55)(21,54)(22,53)(23,44)(24,46)
(25,45)(26,49)(27,48)(28,47)(29,42)(30,41)(31,43)(59,61)(62,63)(66,67)(68,83)
(69,85)(70,84)(71,79)(72,78)(73,77)(74,81)(75,80)(76,82);
s4 := Sym(85)!( 5,14)( 6,16)( 7,15)( 8,17)( 9,19)(10,18)(11,20)(12,22)(13,21)
(24,25)(27,28)(30,31)(32,70)(33,69)(34,68)(35,73)(36,72)(37,71)(38,76)(39,75)
(40,74)(41,61)(42,60)(43,59)(44,64)(45,63)(46,62)(47,67)(48,66)(49,65)(50,79)
(51,78)(52,77)(53,82)(54,81)(55,80)(56,85)(57,84)(58,83);
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, 
s3*s4*s3*s4*s3*s4, s2*s4*s3*s2*s4*s3*s2*s4*s3*s2*s4*s3*s2*s4*s3*s2*s4*s3, 
s4*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s4*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >;

to this polytope