Polytope of Type {2,18,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,18,4}*1296
if this polytope has a name.
Group : SmallGroup(1296,1813)
Rank : 4
Schlafli Type : {2,18,4}
Number of vertices, edges, etc : 2, 81, 162, 18
Order of s₀s₁s₂s₃ : 4
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Degenerate
   Universal
   Non-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) :
   9-fold quotients : {2,6,4}*144
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

to this polytope