Polytope of Type {2,18,3}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,18,3}*648
if this polytope has a name.
Group : SmallGroup(648,301)
Rank : 4
Schlafli Type : {2,18,3}
Number of vertices, edges, etc : 2, 54, 81, 9
Order of s₀s₁s₂s₃ : 6
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Degenerate
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {2,18,3,2} of size 1296
Vertex Figure Of :
   {2,2,18,3} of size 1296
   {3,2,18,3} of size 1944
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {2,6,3}*216
   9-fold quotients : {2,6,3}*72
   27-fold quotients : {2,2,3}*24
Covers (Minimal Covers in Boldface) :
   2-fold covers : {4,18,3}*1296, {2,18,6}*1296h
   3-fold covers : {2,18,3}*1944a, {2,18,9}*1944c, {2,18,9}*1944d, {2,18,9}*1944i, {2,18,9}*1944j, {2,18,3}*1944b, {6,18,3}*1944
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

to this polytope