Polytope of Type {2,6,9}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,6,9}*648c
if this polytope has a name.
Group : SmallGroup(648,299)
Rank : 4
Schlafli Type : {2,6,9}
Number of vertices, edges, etc : 2, 18, 81, 27
Order of s₀s₁s₂s₃ : 6
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Degenerate
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {2,6,9,2} of size 1296
Vertex Figure Of :
   {2,2,6,9} of size 1296
   {3,2,6,9} of size 1944
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {2,6,3}*216
   9-fold quotients : {2,6,3}*72
   27-fold quotients : {2,2,3}*24
Covers (Minimal Covers in Boldface) :
   2-fold covers : {4,6,9}*1296b, {2,6,18}*1296c
   3-fold covers : {2,6,9}*1944a, {2,18,9}*1944b, {2,18,9}*1944c, {2,18,9}*1944f, {2,6,9}*1944e, {6,6,9}*1944f
Permutation Representation (GAP) :

s0 := (1,2);;
s1 := ( 3, 9)( 4,10)( 5,11);;
s2 := ( 4, 5)( 6, 9)( 7,11)( 8,10);;
s3 := ( 3, 9)( 4,11)( 5,10)( 6, 7);;
poly := Group([s0,s1,s2,s3]);;

Finitely Presented Group Representation (GAP) :

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

Permutation Representation (Magma) :

s0 := Sym(11)!(1,2);
s1 := Sym(11)!( 3, 9)( 4,10)( 5,11);
s2 := Sym(11)!( 4, 5)( 6, 9)( 7,11)( 8,10);
s3 := Sym(11)!( 3, 9)( 4,11)( 5,10)( 6, 7);
poly := sub<Sym(11)|s0,s1,s2,s3>;

Finitely Presented Group Representation (Magma) :

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

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