Polytope of Type {2,9,6}

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

s0 := (1,2);;
s1 := ( 4, 5)( 6, 7)( 8,11)( 9,13)(10,12)(14,17)(15,19)(16,18)(20,23)(21,25)
(22,24)(26,29)(27,28);;
s2 := ( 3, 9)( 4, 6)( 5,15)( 7,10)( 8,12)(11,21)(13,16)(14,18)(17,26)(19,22)
(20,24)(23,28)(25,27);;
s3 := ( 6, 7)( 9,10)(12,13)(15,16)(18,19)(21,22)(24,25)(26,27)(28,29);;
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, 
s3*s1*s2*s3*s2*s3*s1*s2*s3*s2, s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(29)!(1,2);
s1 := Sym(29)!( 4, 5)( 6, 7)( 8,11)( 9,13)(10,12)(14,17)(15,19)(16,18)(20,23)
(21,25)(22,24)(26,29)(27,28);
s2 := Sym(29)!( 3, 9)( 4, 6)( 5,15)( 7,10)( 8,12)(11,21)(13,16)(14,18)(17,26)
(19,22)(20,24)(23,28)(25,27);
s3 := Sym(29)!( 6, 7)( 9,10)(12,13)(15,16)(18,19)(21,22)(24,25)(26,27)(28,29);
poly := sub<Sym(29)|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, s3*s1*s2*s3*s2*s3*s1*s2*s3*s2, 
s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;

to this polytope