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

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

Permutation Representation (Magma) :

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

to this polytope