Polytope of Type {3,2,6,18}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,2,6,18}*1944a
if this polytope has a name.
Group : SmallGroup(1944,2340)
Rank : 5
Schlafli Type : {3,2,6,18}
Number of vertices, edges, etc : 3, 3, 9, 81, 27
Order of s₀s₁s₂s₃s₄ : 9
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 : {3,2,6,6}*648
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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