Polytope of Type {6,6,3}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,6,3}*216a
if this polytope has a name.
Group : SmallGroup(216,102)
Rank : 4
Schlafli Type : {6,6,3}
Number of vertices, edges, etc : 6, 18, 9, 3
Order of s₀s₁s₂s₃ : 6
Order of s₀s₁s₂s₃s₂s₁ : 6
Special Properties :
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {6,6,3,2} of size 432
   {6,6,3,4} of size 864
   {6,6,3,6} of size 1296
   {6,6,3,4} of size 1728
Vertex Figure Of :
   {2,6,6,3} of size 432
   {4,6,6,3} of size 864
   {4,6,6,3} of size 864
   {4,6,6,3} of size 864
   {6,6,6,3} of size 1296
   {6,6,6,3} of size 1296
   {8,6,6,3} of size 1728
   {4,6,6,3} of size 1728
   {6,6,6,3} of size 1944
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {3,6,3}*108
   3-fold quotients : {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 : {12,6,3}*432a, {6,6,6}*432a
   3-fold covers : {6,6,9}*648a, {18,6,3}*648a, {6,6,3}*648a, {6,6,3}*648b, {6,6,3}*648e
   4-fold covers : {24,6,3}*864a, {6,6,12}*864a, {12,6,6}*864a, {6,12,6}*864a, {6,12,3}*864a
   5-fold covers : {6,6,15}*1080a, {30,6,3}*1080a
   6-fold covers : {12,6,9}*1296a, {36,6,3}*1296a, {12,6,3}*1296a, {12,6,3}*1296b, {6,6,18}*1296a, {18,6,6}*1296a, {6,6,6}*1296a, {6,6,6}*1296b, {12,6,3}*1296c, {6,6,6}*1296n, {6,6,6}*1296p
   7-fold covers : {6,6,21}*1512a, {42,6,3}*1512a
   8-fold covers : {48,6,3}*1728a, {12,6,12}*1728a, {6,12,12}*1728a, {12,12,6}*1728a, {6,6,24}*1728a, {24,6,6}*1728a, {6,24,6}*1728a, {12,12,3}*1728a, {6,24,3}*1728a, {6,12,6}*1728a, {6,12,6}*1728b
   9-fold covers : {18,6,9}*1944a, {6,6,3}*1944a, {6,6,27}*1944a, {54,6,3}*1944a, {6,6,9}*1944a, {18,6,3}*1944a, {6,6,9}*1944b, {18,6,3}*1944b, {6,6,9}*1944d, {18,6,3}*1944d, {6,6,3}*1944c, {6,6,3}*1944d, {6,6,3}*1944e, {6,6,3}*1944g
Permutation Representation (GAP) :

s0 := ( 2, 3)( 5, 6)( 8, 9)(11,12)(14,15)(17,18);;
s1 := ( 1,10)( 2,12)( 3,11)( 4,14)( 5,13)( 6,15)( 7,18)( 8,17)( 9,16);;
s2 := ( 1, 4)( 2, 6)( 3, 5)( 8, 9)(10,13)(11,15)(12,14)(17,18);;
s3 := ( 2, 3)( 4, 7)( 5, 9)( 6, 8)(11,12)(13,16)(14,18)(15,17);;
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*s3, 
s2*s0*s1*s2*s1*s2*s0*s1*s2*s1, s3*s1*s2*s1*s2*s3*s1*s2*s1*s2, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(18)!( 2, 3)( 5, 6)( 8, 9)(11,12)(14,15)(17,18);
s1 := Sym(18)!( 1,10)( 2,12)( 3,11)( 4,14)( 5,13)( 6,15)( 7,18)( 8,17)( 9,16);
s2 := Sym(18)!( 1, 4)( 2, 6)( 3, 5)( 8, 9)(10,13)(11,15)(12,14)(17,18);
s3 := Sym(18)!( 2, 3)( 4, 7)( 5, 9)( 6, 8)(11,12)(13,16)(14,18)(15,17);
poly := sub<Sym(18)|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*s3, s2*s0*s1*s2*s1*s2*s0*s1*s2*s1, 
s3*s1*s2*s1*s2*s3*s1*s2*s1*s2, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >;

References : None.
to this polytope