Polytope of Type {2,2,18,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,2,18,6}*1296a
if this polytope has a name.
Group : SmallGroup(1296,1858)
Rank : 5
Schlafli Type : {2,2,18,6}
Number of vertices, edges, etc : 2, 2, 27, 81, 9
Order of s₀s₁s₂s₃s₄ : 18
Order of s₀s₁s₂s₃s₄s₃s₂s₁ : 2
Special Properties :
   Degenerate
   Universal
   Non-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 : {2,2,6,6}*432
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)(32,62)(33,64)(34,63)(35,59)(36,61)(37,60)(38,65)(39,67)(40,66)
(41,71)(42,73)(43,72)(44,68)(45,70)(46,69)(47,74)(48,76)(49,75)(50,80)(51,82)
(52,81)(53,77)(54,79)(55,78)(56,83)(57,85)(58,84);;
s3 := ( 5,32)( 6,33)( 7,34)( 8,38)( 9,39)(10,40)(11,35)(12,36)(13,37)(14,52)
(15,50)(16,51)(17,58)(18,56)(19,57)(20,55)(21,53)(22,54)(23,42)(24,43)(25,41)
(26,48)(27,49)(28,47)(29,45)(30,46)(31,44)(59,62)(60,63)(61,64)(68,82)(69,80)
(70,81)(71,79)(72,77)(73,78)(74,85)(75,83)(76,84);;
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)(32,41)(33,43)(34,42)(35,44)(36,46)(37,45)(38,47)(39,49)(40,48)
(51,52)(54,55)(57,58)(59,68)(60,70)(61,69)(62,71)(63,73)(64,72)(65,74)(66,76)
(67,75)(78,79)(81,82)(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, 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*s3*s4*s3*s4*s3*s4, 
s2*s3*s4*s3*s2*s3*s4*s3*s4*s3*s2*s3*s4*s3*s2*s3, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s4*s2*s3*s4*s2*s3*s4*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)(32,62)(33,64)(34,63)(35,59)(36,61)(37,60)(38,65)(39,67)
(40,66)(41,71)(42,73)(43,72)(44,68)(45,70)(46,69)(47,74)(48,76)(49,75)(50,80)
(51,82)(52,81)(53,77)(54,79)(55,78)(56,83)(57,85)(58,84);
s3 := Sym(85)!( 5,32)( 6,33)( 7,34)( 8,38)( 9,39)(10,40)(11,35)(12,36)(13,37)
(14,52)(15,50)(16,51)(17,58)(18,56)(19,57)(20,55)(21,53)(22,54)(23,42)(24,43)
(25,41)(26,48)(27,49)(28,47)(29,45)(30,46)(31,44)(59,62)(60,63)(61,64)(68,82)
(69,80)(70,81)(71,79)(72,77)(73,78)(74,85)(75,83)(76,84);
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)(32,41)(33,43)(34,42)(35,44)(36,46)(37,45)(38,47)(39,49)
(40,48)(51,52)(54,55)(57,58)(59,68)(60,70)(61,69)(62,71)(63,73)(64,72)(65,74)
(66,76)(67,75)(78,79)(81,82)(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, 
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*s3*s4*s3*s4*s3*s4, 
s2*s3*s4*s3*s2*s3*s4*s3*s4*s3*s2*s3*s4*s3*s2*s3, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s4*s2*s3*s4*s2*s3*s4*s2*s3*s2*s3 >;

to this polytope