Polytope of Type {6,9,2}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,9,2}*216
if this polytope has a name.
Group : SmallGroup(216,101)
Rank : 4
Schlafli Type : {6,9,2}
Number of vertices, edges, etc : 6, 27, 9, 2
Order of s₀s₁s₂s₃ : 18
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Degenerate
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {6,9,2,2} of size 432
   {6,9,2,3} of size 648
   {6,9,2,4} of size 864
   {6,9,2,5} of size 1080
   {6,9,2,6} of size 1296
   {6,9,2,7} of size 1512
   {6,9,2,8} of size 1728
   {6,9,2,9} of size 1944
Vertex Figure Of :
   {2,6,9,2} of size 432
   {3,6,9,2} of size 648
   {4,6,9,2} of size 864
   {6,6,9,2} of size 1296
   {6,6,9,2} of size 1296
   {8,6,9,2} of size 1728
   {9,6,9,2} of size 1944
   {3,6,9,2} of size 1944
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {2,9,2}*72, {6,3,2}*72
   9-fold quotients : {2,3,2}*24
Covers (Minimal Covers in Boldface) :
   2-fold covers : {6,18,2}*432b
   3-fold covers : {18,9,2}*648, {6,9,2}*648a, {6,27,2}*648, {6,9,6}*648
   4-fold covers : {6,36,2}*864b, {6,18,4}*864b, {12,18,2}*864b, {6,9,2}*864, {6,9,4}*864, {12,9,2}*864
   5-fold covers : {6,45,2}*1080
   6-fold covers : {18,18,2}*1296b, {6,18,2}*1296a, {6,54,2}*1296b, {6,18,6}*1296b, {6,18,6}*1296d, {6,18,2}*1296i
   7-fold covers : {6,63,2}*1512
   8-fold covers : {6,36,4}*1728b, {6,72,2}*1728b, {6,18,8}*1728b, {12,36,2}*1728b, {24,18,2}*1728b, {12,18,4}*1728b, {12,9,2}*1728, {24,9,2}*1728, {6,9,8}*1728, {6,18,2}*1728, {6,18,4}*1728b, {12,18,2}*1728b
   9-fold covers : {18,9,2}*1944a, {18,27,2}*1944, {6,27,2}*1944a, {6,9,2}*1944d, {18,9,2}*1944h, {18,9,2}*1944i, {6,9,2}*1944e, {6,27,2}*1944b, {6,27,2}*1944c, {6,81,2}*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 := ( 4, 5)( 7, 8)(10,11)(13,14)(16,17)(19,20)(22,23)(24,25)(26,27);;
s1 := ( 1, 4)( 2,10)( 3, 7)( 6,16)( 8,11)( 9,13)(12,22)(14,17)(15,19)(18,26)
(20,23)(21,24)(25,27);;
s2 := ( 1, 2)( 3, 6)( 4, 8)( 5, 7)( 9,12)(10,14)(11,13)(15,18)(16,20)(17,19)
(22,25)(23,24)(26,27);;
s3 := (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*s2*s0*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s2*s3*s2*s3, 
s2*s0*s1*s0*s1*s2*s0*s1*s0*s1, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
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)!( 4, 5)( 7, 8)(10,11)(13,14)(16,17)(19,20)(22,23)(24,25)(26,27);
s1 := Sym(29)!( 1, 4)( 2,10)( 3, 7)( 6,16)( 8,11)( 9,13)(12,22)(14,17)(15,19)
(18,26)(20,23)(21,24)(25,27);
s2 := Sym(29)!( 1, 2)( 3, 6)( 4, 8)( 5, 7)( 9,12)(10,14)(11,13)(15,18)(16,20)
(17,19)(22,25)(23,24)(26,27);
s3 := Sym(29)!(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*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s2*s3*s2*s3, s2*s0*s1*s0*s1*s2*s0*s1*s0*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;

to this polytope