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

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,2,18,6}*1296a
if this polytope has a name.
Group : SmallGroup(1296,2984)
Rank : 5
Schlafli Type : {3,2,18,6}
Number of vertices, edges, etc : 3, 3, 18, 54, 6
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 : {3,2,18,2}*432, {3,2,6,6}*432a
   6-fold quotients : {3,2,9,2}*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 := ( 5, 6)( 8, 9)(11,12)(13,23)(14,22)(15,24)(16,26)(17,25)(18,27)(19,29)
(20,28)(21,30)(32,33)(35,36)(38,39)(40,50)(41,49)(42,51)(43,53)(44,52)(45,54)
(46,56)(47,55)(48,57);;
s3 := ( 4,13)( 5,15)( 6,14)( 7,19)( 8,21)( 9,20)(10,16)(11,18)(12,17)(22,23)
(25,29)(26,28)(27,30)(31,40)(32,42)(33,41)(34,46)(35,48)(36,47)(37,43)(38,45)
(39,44)(49,50)(52,56)(53,55)(54,57);;
s4 := ( 4,34)( 5,35)( 6,36)( 7,31)( 8,32)( 9,33)(10,37)(11,38)(12,39)(13,43)
(14,44)(15,45)(16,40)(17,41)(18,42)(19,46)(20,47)(21,48)(22,52)(23,53)(24,54)
(25,49)(26,50)(27,51)(28,55)(29,56)(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, 
s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*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(57)!(2,3);
s1 := Sym(57)!(1,2);
s2 := Sym(57)!( 5, 6)( 8, 9)(11,12)(13,23)(14,22)(15,24)(16,26)(17,25)(18,27)
(19,29)(20,28)(21,30)(32,33)(35,36)(38,39)(40,50)(41,49)(42,51)(43,53)(44,52)
(45,54)(46,56)(47,55)(48,57);
s3 := Sym(57)!( 4,13)( 5,15)( 6,14)( 7,19)( 8,21)( 9,20)(10,16)(11,18)(12,17)
(22,23)(25,29)(26,28)(27,30)(31,40)(32,42)(33,41)(34,46)(35,48)(36,47)(37,43)
(38,45)(39,44)(49,50)(52,56)(53,55)(54,57);
s4 := Sym(57)!( 4,34)( 5,35)( 6,36)( 7,31)( 8,32)( 9,33)(10,37)(11,38)(12,39)
(13,43)(14,44)(15,45)(16,40)(17,41)(18,42)(19,46)(20,47)(21,48)(22,52)(23,53)
(24,54)(25,49)(26,50)(27,51)(28,55)(29,56)(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, s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >;

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