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

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

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