Polytope of Type {3,2,6,18}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,2,6,18}*1296a
if this polytope has a name.
Group : SmallGroup(1296,2984)
Rank : 5
Schlafli Type : {3,2,6,18}
Number of vertices, edges, etc : 3, 3, 6, 54, 18
Order of s₀s₁s₂s₃s₄ : 18
Order of s₀s₁s₂s₃s₄s₃s₂s₁ : 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 : {3,2,2,18}*432, {3,2,6,6}*432a
   6-fold quotients : {3,2,2,9}*216
   9-fold quotients : {3,2,2,6}*144, {3,2,6,2}*144
   18-fold quotients : {3,2,2,3}*72, {3,2,3,2}*72
   27-fold quotients : {3,2,2,2}*48
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := (2,3);;
s1 := (1,2);;
s2 := ( 7,10)( 8,11)( 9,12)(16,19)(17,20)(18,21)(25,28)(26,29)(27,30)(34,37)
(35,38)(36,39)(43,46)(44,47)(45,48)(52,55)(53,56)(54,57);;
s3 := ( 4, 7)( 5, 9)( 6, 8)(11,12)(13,26)(14,25)(15,27)(16,23)(17,22)(18,24)
(19,29)(20,28)(21,30)(31,34)(32,36)(33,35)(38,39)(40,53)(41,52)(42,54)(43,50)
(44,49)(45,51)(46,56)(47,55)(48,57);;
s4 := ( 4,40)( 5,42)( 6,41)( 7,43)( 8,45)( 9,44)(10,46)(11,48)(12,47)(13,31)
(14,33)(15,32)(16,34)(17,36)(18,35)(19,37)(20,39)(21,38)(22,50)(23,49)(24,51)
(25,53)(26,52)(27,54)(28,56)(29,55)(30,57);;
poly := Group([s0,s1,s2,s3,s4]);;

Finitely Presented Group Representation (GAP) :

F := FreeGroup("s0","s1","s2","s3","s4");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  s4 := F.5;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s4*s4, s0*s2*s0*s2, 
s1*s2*s1*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s0*s4*s0*s4, s1*s4*s1*s4, s2*s4*s2*s4, 
s0*s1*s0*s1*s0*s1, s2*s3*s4*s3*s2*s3*s4*s3, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(57)!(2,3);
s1 := Sym(57)!(1,2);
s2 := Sym(57)!( 7,10)( 8,11)( 9,12)(16,19)(17,20)(18,21)(25,28)(26,29)(27,30)
(34,37)(35,38)(36,39)(43,46)(44,47)(45,48)(52,55)(53,56)(54,57);
s3 := Sym(57)!( 4, 7)( 5, 9)( 6, 8)(11,12)(13,26)(14,25)(15,27)(16,23)(17,22)
(18,24)(19,29)(20,28)(21,30)(31,34)(32,36)(33,35)(38,39)(40,53)(41,52)(42,54)
(43,50)(44,49)(45,51)(46,56)(47,55)(48,57);
s4 := Sym(57)!( 4,40)( 5,42)( 6,41)( 7,43)( 8,45)( 9,44)(10,46)(11,48)(12,47)
(13,31)(14,33)(15,32)(16,34)(17,36)(18,35)(19,37)(20,39)(21,38)(22,50)(23,49)
(24,51)(25,53)(26,52)(27,54)(28,56)(29,55)(30,57);
poly := sub<Sym(57)|s0,s1,s2,s3,s4>;

Finitely Presented Group Representation (Magma) :

poly<s0,s1,s2,s3,s4> := Group< s0,s1,s2,s3,s4 | s0*s0, s1*s1, s2*s2, 
s3*s3, s4*s4, s0*s2*s0*s2, s1*s2*s1*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s0*s4*s0*s4, 
s1*s4*s1*s4, s2*s4*s2*s4, s0*s1*s0*s1*s0*s1, 
s2*s3*s4*s3*s2*s3*s4*s3, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4 >;

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