Polytope of Type {6,2,4,9}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,2,4,9}*864
if this polytope has a name.
Group : SmallGroup(864,3999)
Rank : 5
Schlafli Type : {6,2,4,9}
Number of vertices, edges, etc : 6, 6, 4, 18, 9
Order of s₀s₁s₂s₃s₄ : 18
Order of s₀s₁s₂s₃s₄s₃s₂s₁ : 2
Special Properties :
   Degenerate
   Universal
   Non-Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {6,2,4,9,2} of size 1728
Vertex Figure Of :
   {2,6,2,4,9} of size 1728
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {3,2,4,9}*432
   3-fold quotients : {2,2,4,9}*288, {6,2,4,3}*288
   6-fold quotients : {3,2,4,3}*144
   9-fold quotients : {2,2,4,3}*96
Covers (Minimal Covers in Boldface) :
   2-fold covers : {12,2,4,9}*1728, {6,2,4,9}*1728, {6,2,4,18}*1728b, {6,2,4,18}*1728c
Permutation Representation (GAP) :

s0 := (3,4)(5,6);;
s1 := (1,5)(2,3)(4,6);;
s2 := ( 8,13)( 9,15)(10,17)(11,19)(14,24)(16,26)(20,30)(27,36)(29,38)(31,39)
(33,40)(35,41);;
s3 := ( 7, 8)( 9,12)(10,11)(13,21)(14,20)(15,22)(16,18)(17,19)(23,29)(24,30)
(25,27)(26,28)(31,37)(32,38)(33,35)(34,36)(39,42)(40,41);;
s4 := ( 7,12)( 8,10)( 9,20)(11,16)(13,17)(14,29)(15,30)(18,25)(19,26)(21,22)
(23,37)(24,38)(27,33)(28,34)(31,35)(32,42)(36,40)(39,41);;
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, 
s2*s3*s2*s3*s2*s3*s2*s3, s2*s3*s4*s3*s2*s3*s4*s2*s3, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
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(42)!(3,4)(5,6);
s1 := Sym(42)!(1,5)(2,3)(4,6);
s2 := Sym(42)!( 8,13)( 9,15)(10,17)(11,19)(14,24)(16,26)(20,30)(27,36)(29,38)
(31,39)(33,40)(35,41);
s3 := Sym(42)!( 7, 8)( 9,12)(10,11)(13,21)(14,20)(15,22)(16,18)(17,19)(23,29)
(24,30)(25,27)(26,28)(31,37)(32,38)(33,35)(34,36)(39,42)(40,41);
s4 := Sym(42)!( 7,12)( 8,10)( 9,20)(11,16)(13,17)(14,29)(15,30)(18,25)(19,26)
(21,22)(23,37)(24,38)(27,33)(28,34)(31,35)(32,42)(36,40)(39,41);
poly := sub<Sym(42)|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, s2*s3*s2*s3*s2*s3*s2*s3, 
s2*s3*s4*s3*s2*s3*s4*s2*s3, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4 >;

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