Polytope of Type {3,2,18,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,2,18,4}*864a
if this polytope has a name.
Group : SmallGroup(864,2462)
Rank : 5
Schlafli Type : {3,2,18,4}
Number of vertices, edges, etc : 3, 3, 18, 36, 4
Order of s₀s₁s₂s₃s₄ : 36
Order of s₀s₁s₂s₃s₄s₃s₂s₁ : 2
Special Properties :
   Degenerate
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {3,2,18,4,2} of size 1728
Vertex Figure Of :
   {2,3,2,18,4} of size 1728
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {3,2,18,2}*432
   3-fold quotients : {3,2,6,4}*288a
   4-fold quotients : {3,2,9,2}*216
   6-fold quotients : {3,2,6,2}*144
   9-fold quotients : {3,2,2,4}*96
   12-fold quotients : {3,2,3,2}*72
   18-fold quotients : {3,2,2,2}*48
Covers (Minimal Covers in Boldface) :
   2-fold covers : {3,2,36,4}*1728a, {3,2,18,8}*1728, {6,2,18,4}*1728a
Permutation Representation (GAP) :

s0 := (2,3);;
s1 := (1,2);;
s2 := ( 5, 6)( 7,11)( 8,10)( 9,12)(14,15)(16,20)(17,19)(18,21)(23,24)(25,29)
(26,28)(27,30)(32,33)(34,38)(35,37)(36,39);;
s3 := ( 4, 7)( 5, 9)( 6, 8)(10,11)(13,16)(14,18)(15,17)(19,20)(22,34)(23,36)
(24,35)(25,31)(26,33)(27,32)(28,38)(29,37)(30,39);;
s4 := ( 4,22)( 5,23)( 6,24)( 7,25)( 8,26)( 9,27)(10,28)(11,29)(12,30)(13,31)
(14,32)(15,33)(16,34)(17,35)(18,36)(19,37)(20,38)(21,39);;
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, 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(39)!(2,3);
s1 := Sym(39)!(1,2);
s2 := Sym(39)!( 5, 6)( 7,11)( 8,10)( 9,12)(14,15)(16,20)(17,19)(18,21)(23,24)
(25,29)(26,28)(27,30)(32,33)(34,38)(35,37)(36,39);
s3 := Sym(39)!( 4, 7)( 5, 9)( 6, 8)(10,11)(13,16)(14,18)(15,17)(19,20)(22,34)
(23,36)(24,35)(25,31)(26,33)(27,32)(28,38)(29,37)(30,39);
s4 := Sym(39)!( 4,22)( 5,23)( 6,24)( 7,25)( 8,26)( 9,27)(10,28)(11,29)(12,30)
(13,31)(14,32)(15,33)(16,34)(17,35)(18,36)(19,37)(20,38)(21,39);
poly := sub<Sym(39)|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, 
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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