Polytope of Type {2,9,18}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,9,18}*1944c
if this polytope has a name.
Group : SmallGroup(1944,944)
Rank : 4
Schlafli Type : {2,9,18}
Number of vertices, edges, etc : 2, 27, 243, 54
Order of s₀s₁s₂s₃ : 6
Order of s₀s₁s₂s₃s₂s₁ : 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 : {2,9,6}*648c, {2,3,18}*648
   9-fold quotients : {2,3,6}*216
   27-fold quotients : {2,3,6}*72
   81-fold quotients : {2,3,2}*24
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

Finitely Presented Group Representation (GAP) :

F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s1*s0*s1, 
s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2, 
s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s3*s1*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s1*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2 ];;
poly := F / rels;;

Permutation Representation (Magma) :

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

Finitely Presented Group Representation (Magma) :

poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2, 
s3*s3, s0*s1*s0*s1, s0*s2*s0*s2, s0*s3*s0*s3, 
s1*s3*s1*s3, s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2, 
s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s3*s1*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s1*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2 >;

to this polytope