Polytope of Type {3,2,3,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,2,3,6}*216
if this polytope has a name.
Group : SmallGroup(216,162)
Rank : 5
Schlafli Type : {3,2,3,6}
Number of vertices, edges, etc : 3, 3, 3, 9, 6
Order of s₀s₁s₂s₃s₄ : 6
Order of s₀s₁s₂s₃s₄s₃s₂s₁ : 2
Special Properties :
   Degenerate
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {3,2,3,6,2} of size 432
   {3,2,3,6,3} of size 648
   {3,2,3,6,4} of size 864
   {3,2,3,6,6} of size 1296
   {3,2,3,6,6} of size 1296
   {3,2,3,6,8} of size 1728
   {3,2,3,6,9} of size 1944
   {3,2,3,6,3} of size 1944
Vertex Figure Of :
   {2,3,2,3,6} of size 432
   {3,3,2,3,6} of size 864
   {4,3,2,3,6} of size 864
   {6,3,2,3,6} of size 1296
   {4,3,2,3,6} of size 1728
   {6,3,2,3,6} of size 1728
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {3,2,3,2}*72
Covers (Minimal Covers in Boldface) :
   2-fold covers : {3,2,6,6}*432c, {6,2,3,6}*432
   3-fold covers : {3,2,9,6}*648, {9,2,3,6}*648, {3,6,3,6}*648, {3,2,3,6}*648
   4-fold covers : {3,2,12,6}*864b, {12,2,3,6}*864, {3,2,6,12}*864c, {3,2,3,6}*864, {3,2,3,12}*864, {6,2,6,6}*864c
   5-fold covers : {3,2,15,6}*1080, {15,2,3,6}*1080
   6-fold covers : {3,2,18,6}*1296b, {6,2,9,6}*1296, {9,2,6,6}*1296c, {18,2,3,6}*1296, {3,6,6,6}*1296b, {6,6,3,6}*1296a, {3,2,6,6}*1296c, {6,2,3,6}*1296, {3,6,6,6}*1296d, {6,6,3,6}*1296b, {3,2,6,6}*1296d
   7-fold covers : {3,2,21,6}*1512, {21,2,3,6}*1512
   8-fold covers : {3,2,24,6}*1728b, {24,2,3,6}*1728, {3,2,12,12}*1728c, {3,2,6,24}*1728c, {3,2,3,12}*1728, {3,2,3,24}*1728, {6,2,12,6}*1728b, {12,2,6,6}*1728c, {6,4,6,6}*1728b, {6,2,6,12}*1728c, {3,4,6,6}*1728c, {6,4,3,6}*1728, {3,2,6,6}*1728b, {3,2,6,12}*1728b, {6,2,3,6}*1728, {6,2,3,12}*1728
   9-fold covers : {3,2,9,18}*1944, {9,2,9,6}*1944, {3,6,9,6}*1944, {3,2,9,6}*1944a, {9,6,3,6}*1944, {9,2,3,6}*1944, {3,6,3,6}*1944a, {3,2,27,6}*1944, {27,2,3,6}*1944, {3,6,3,6}*1944b, {3,6,3,6}*1944c, {3,2,9,6}*1944b, {3,2,9,6}*1944c, {3,2,9,6}*1944d, {3,2,3,6}*1944, {3,2,3,18}*1944
Permutation Representation (GAP) :

s0 := (2,3);;
s1 := (1,2);;
s2 := ( 5, 6)( 7, 8)( 9,12)(10,11);;
s3 := ( 4, 9)( 5, 7)( 6,11)( 8,10);;
s4 := ( 7, 8)( 9,10)(11,12);;
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*s2*s0*s2, 
s1*s2*s1*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s0*s4*s0*s4, s1*s4*s1*s4, s2*s4*s2*s4, 
s0*s1*s0*s1*s0*s1, s2*s3*s2*s3*s2*s3, 
s4*s2*s3*s4*s3*s4*s2*s3*s4*s3 ];;
poly := F / rels;;

Permutation Representation (Magma) :

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

to this polytope