Polytope of Type {9,2,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {9,2,6}*216
if this polytope has a name.
Group : SmallGroup(216,101)
Rank : 4
Schlafli Type : {9,2,6}
Number of vertices, edges, etc : 9, 9, 6, 6
Order of s₀s₁s₂s₃ : 18
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Degenerate
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {9,2,6,2} of size 432
   {9,2,6,3} of size 648
   {9,2,6,4} of size 864
   {9,2,6,3} of size 864
   {9,2,6,4} of size 864
   {9,2,6,4} of size 864
   {9,2,6,4} of size 1296
   {9,2,6,6} of size 1296
   {9,2,6,6} of size 1296
   {9,2,6,6} of size 1296
   {9,2,6,8} of size 1728
   {9,2,6,4} of size 1728
   {9,2,6,6} of size 1728
   {9,2,6,9} of size 1944
   {9,2,6,3} of size 1944
   {9,2,6,6} of size 1944
Vertex Figure Of :
   {2,9,2,6} of size 432
   {4,9,2,6} of size 864
   {6,9,2,6} of size 1296
   {4,9,2,6} of size 1728
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {9,2,3}*108
   3-fold quotients : {9,2,2}*72, {3,2,6}*72
   6-fold quotients : {3,2,3}*36
   9-fold quotients : {3,2,2}*24
Covers (Minimal Covers in Boldface) :
   2-fold covers : {9,2,12}*432, {18,2,6}*432
   3-fold covers : {9,2,18}*648, {9,6,6}*648a, {27,2,6}*648, {9,6,6}*648b
   4-fold covers : {9,2,24}*864, {36,2,6}*864, {18,2,12}*864, {18,4,6}*864, {9,4,6}*864
   5-fold covers : {45,2,6}*1080, {9,2,30}*1080
   6-fold covers : {9,2,36}*1296, {9,6,12}*1296a, {27,2,12}*1296, {18,2,18}*1296, {18,6,6}*1296a, {54,2,6}*1296, {9,6,12}*1296b, {18,6,6}*1296b, {18,6,6}*1296c, {18,6,6}*1296e
   7-fold covers : {63,2,6}*1512, {9,2,42}*1512
   8-fold covers : {9,2,48}*1728, {36,2,12}*1728, {18,4,12}*1728, {36,4,6}*1728, {72,2,6}*1728, {18,2,24}*1728, {18,8,6}*1728, {9,4,12}*1728, {9,8,6}*1728, {18,4,6}*1728a, {18,4,6}*1728b
   9-fold covers : {9,6,18}*1944a, {9,2,54}*1944, {27,2,18}*1944, {27,6,6}*1944a, {9,6,6}*1944a, {9,6,6}*1944b, {81,2,6}*1944, {9,6,18}*1944b, {9,18,6}*1944, {9,6,6}*1944c, {9,6,6}*1944d, {9,6,6}*1944e, {27,6,6}*1944b
Permutation Representation (GAP) :

s0 := (2,3)(4,5)(6,7)(8,9);;
s1 := (1,2)(3,4)(5,6)(7,8);;
s2 := (12,13)(14,15);;
s3 := (10,14)(11,12)(13,15);;
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*s2*s0*s2, 
s1*s2*s1*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(15)!(2,3)(4,5)(6,7)(8,9);
s1 := Sym(15)!(1,2)(3,4)(5,6)(7,8);
s2 := Sym(15)!(12,13)(14,15);
s3 := Sym(15)!(10,14)(11,12)(13,15);
poly := sub<Sym(15)|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*s2*s0*s2, s1*s2*s1*s2, s0*s3*s0*s3, 
s1*s3*s1*s3, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >;

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