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

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,6,18,2}*1296b
if this polytope has a name.
Group : SmallGroup(1296,1861)
Rank : 5
Schlafli Type : {2,6,18,2}
Number of vertices, edges, etc : 2, 9, 81, 27, 2
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 : {2,6,6,2}*432
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

to this polytope