Polytope of Type {2,18,9}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,18,9}*648
if this polytope has a name.
Group : SmallGroup(648,296)
Rank : 4
Schlafli Type : {2,18,9}
Number of vertices, edges, etc : 2, 18, 81, 9
Order of s₀s₁s₂s₃ : 18
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Degenerate
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {2,18,9,2} of size 1296
Vertex Figure Of :
   {2,2,18,9} of size 1296
   {3,2,18,9} of size 1944
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {2,6,9}*216
   9-fold quotients : {2,2,9}*72, {2,6,3}*72
   27-fold quotients : {2,2,3}*24
Covers (Minimal Covers in Boldface) :
   2-fold covers : {4,18,9}*1296, {2,18,18}*1296b
   3-fold covers : {2,18,9}*1944a, {2,18,27}*1944, {6,18,9}*1944
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

to this polytope