Polytope of Type {6,2,9}

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

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

Permutation Representation (Magma) :

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

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