Polytope of Type {2,18,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,18,6}*648a
if this polytope has a name.
Group : SmallGroup(648,297)
Rank : 4
Schlafli Type : {2,18,6}
Number of vertices, edges, etc : 2, 27, 81, 9
Order of s₀s₁s₂s₃ : 18
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Degenerate
   Universal
   Non-Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {2,18,6,2} of size 1296
Vertex Figure Of :
   {2,2,18,6} of size 1296
   {3,2,18,6} of size 1944
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {2,6,6}*216
Covers (Minimal Covers in Boldface) :
   2-fold covers : {2,18,6}*1296b
   3-fold covers : {2,18,18}*1944a, {2,54,6}*1944a, {2,18,6}*1944c, {2,18,18}*1944f, {2,18,18}*1944h, {2,18,6}*1944d, {2,54,6}*1944b, {2,54,6}*1944c, {6,18,6}*1944a
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)(30,60)(31,62)(32,61)(33,57)(34,59)(35,58)(36,63)(37,65)(38,64)
(39,69)(40,71)(41,70)(42,66)(43,68)(44,67)(45,72)(46,74)(47,73)(48,78)(49,80)
(50,79)(51,75)(52,77)(53,76)(54,81)(55,83)(56,82);;
s2 := ( 3,30)( 4,31)( 5,32)( 6,36)( 7,37)( 8,38)( 9,33)(10,34)(11,35)(12,50)
(13,48)(14,49)(15,56)(16,54)(17,55)(18,53)(19,51)(20,52)(21,40)(22,41)(23,39)
(24,46)(25,47)(26,45)(27,43)(28,44)(29,42)(57,60)(58,61)(59,62)(66,80)(67,78)
(68,79)(69,77)(70,75)(71,76)(72,83)(73,81)(74,82);;
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)(30,39)(31,41)(32,40)(33,42)(34,44)(35,43)(36,45)(37,47)(38,46)
(49,50)(52,53)(55,56)(57,66)(58,68)(59,67)(60,69)(61,71)(62,70)(63,72)(64,74)
(65,73)(76,77)(79,80)(82,83);;
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, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s1*s2*s3*s2*s1*s2*s3*s2*s3*s2*s1*s2*s3*s2*s1*s2, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s3*s1*s2*s3*s1*s2*s3*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)(30,60)(31,62)(32,61)(33,57)(34,59)(35,58)(36,63)(37,65)
(38,64)(39,69)(40,71)(41,70)(42,66)(43,68)(44,67)(45,72)(46,74)(47,73)(48,78)
(49,80)(50,79)(51,75)(52,77)(53,76)(54,81)(55,83)(56,82);
s2 := Sym(83)!( 3,30)( 4,31)( 5,32)( 6,36)( 7,37)( 8,38)( 9,33)(10,34)(11,35)
(12,50)(13,48)(14,49)(15,56)(16,54)(17,55)(18,53)(19,51)(20,52)(21,40)(22,41)
(23,39)(24,46)(25,47)(26,45)(27,43)(28,44)(29,42)(57,60)(58,61)(59,62)(66,80)
(67,78)(68,79)(69,77)(70,75)(71,76)(72,83)(73,81)(74,82);
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)(30,39)(31,41)(32,40)(33,42)(34,44)(35,43)(36,45)(37,47)
(38,46)(49,50)(52,53)(55,56)(57,66)(58,68)(59,67)(60,69)(61,71)(62,70)(63,72)
(64,74)(65,73)(76,77)(79,80)(82,83);
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, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s1*s2*s3*s2*s1*s2*s3*s2*s3*s2*s1*s2*s3*s2*s1*s2, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s1*s2 >;

to this polytope