Polytope of Type {2,3,2,9}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,3,2,9}*216
if this polytope has a name.
Group : SmallGroup(216,101)
Rank : 5
Schlafli Type : {2,3,2,9}
Number of vertices, edges, etc : 2, 3, 3, 9, 9
Order of s₀s₁s₂s₃s₄ : 18
Order of s₀s₁s₂s₃s₄s₃s₂s₁ : 2
Special Properties :
   Degenerate
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {2,3,2,9,2} of size 432
   {2,3,2,9,4} of size 864
   {2,3,2,9,6} of size 1296
   {2,3,2,9,4} of size 1728
Vertex Figure Of :
   {2,2,3,2,9} of size 432
   {3,2,3,2,9} of size 648
   {4,2,3,2,9} of size 864
   {5,2,3,2,9} of size 1080
   {6,2,3,2,9} of size 1296
   {7,2,3,2,9} of size 1512
   {8,2,3,2,9} of size 1728
   {9,2,3,2,9} of size 1944
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {2,3,2,3}*72
Covers (Minimal Covers in Boldface) :
   2-fold covers : {2,3,2,18}*432, {2,6,2,9}*432
   3-fold covers : {2,9,2,9}*648, {2,3,6,9}*648, {2,3,2,27}*648, {6,3,2,9}*648
   4-fold covers : {2,3,2,36}*864, {2,12,2,9}*864, {4,6,2,9}*864a, {4,3,2,9}*864, {2,6,2,18}*864
   5-fold covers : {2,3,2,45}*1080, {2,15,2,9}*1080
   6-fold covers : {2,9,2,18}*1296, {2,18,2,9}*1296, {2,3,6,18}*1296a, {2,6,6,9}*1296a, {2,3,2,54}*1296, {2,6,2,27}*1296, {2,3,6,18}*1296b, {2,6,6,9}*1296b, {6,3,2,18}*1296, {6,6,2,9}*1296a, {6,6,2,9}*1296b
   7-fold covers : {2,3,2,63}*1512, {2,21,2,9}*1512
   8-fold covers : {4,12,2,9}*1728a, {2,3,2,72}*1728, {2,24,2,9}*1728, {8,6,2,9}*1728, {8,3,2,9}*1728, {2,12,2,18}*1728, {2,6,2,36}*1728, {2,6,4,18}*1728, {4,6,2,18}*1728a, {2,3,4,18}*1728, {4,3,2,18}*1728, {4,6,2,9}*1728, {2,6,4,9}*1728
   9-fold covers : {2,9,6,9}*1944, {2,9,2,27}*1944, {2,27,2,9}*1944, {2,3,6,27}*1944, {2,3,6,9}*1944a, {2,3,6,9}*1944b, {2,3,2,81}*1944, {6,9,2,9}*1944, {6,3,6,9}*1944, {6,3,2,9}*1944, {6,3,2,27}*1944
Permutation Representation (GAP) :

s0 := (1,2);;
s1 := (4,5);;
s2 := (3,4);;
s3 := ( 7, 8)( 9,10)(11,12)(13,14);;
s4 := ( 6, 7)( 8, 9)(10,11)(12,13);;
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*s1*s0*s1, 
s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s2*s3*s2*s3, s0*s4*s0*s4, s1*s4*s1*s4, 
s2*s4*s2*s4, s1*s2*s1*s2*s1*s2, 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(14)!(1,2);
s1 := Sym(14)!(4,5);
s2 := Sym(14)!(3,4);
s3 := Sym(14)!( 7, 8)( 9,10)(11,12)(13,14);
s4 := Sym(14)!( 6, 7)( 8, 9)(10,11)(12,13);
poly := sub<Sym(14)|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*s1*s0*s1, s0*s2*s0*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s2*s3*s2*s3, 
s0*s4*s0*s4, s1*s4*s1*s4, s2*s4*s2*s4, 
s1*s2*s1*s2*s1*s2, s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4 >;

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