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

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,2,3,18}*1944
if this polytope has a name.
Group : SmallGroup(1944,2346)
Rank : 5
Schlafli Type : {3,2,3,18}
Number of vertices, edges, etc : 3, 3, 9, 81, 54
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 : {3,2,3,6}*648
   9-fold quotients : {3,2,3,6}*216
   27-fold quotients : {3,2,3,2}*72
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

to this polytope