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Polytope of Type {3,6,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,6,6}*216a
if this polytope has a name.
Group : SmallGroup(216,102)
Rank : 4
Schlafli Type : {3,6,6}
Number of vertices, edges, etc : 3, 9, 18, 6
Order of s0s1s2s3 : 6
Order of s0s1s2s3s2s1 : 6
Special Properties :
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {3,6,6,2} of size 432
   {3,6,6,4} of size 864
   {3,6,6,4} of size 864
   {3,6,6,4} of size 864
   {3,6,6,6} of size 1296
   {3,6,6,6} of size 1296
   {3,6,6,8} of size 1728
   {3,6,6,4} of size 1728
   {3,6,6,6} of size 1944
Vertex Figure Of :
   {2,3,6,6} of size 432
   {4,3,6,6} of size 864
   {6,3,6,6} of size 1296
   {4,3,6,6} of size 1728
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {3,6,3}*108
   3-fold quotients : {3,2,6}*72
   6-fold quotients : {3,2,3}*36
   9-fold quotients : {3,2,2}*24
Covers (Minimal Covers in Boldface) :
   2-fold covers : {3,6,12}*432a, {6,6,6}*432a
   3-fold covers : {3,6,18}*648a, {9,6,6}*648a, {3,6,6}*648a, {3,6,6}*648b, {3,6,6}*648e
   4-fold covers : {3,6,24}*864a, {6,6,12}*864a, {12,6,6}*864a, {6,12,6}*864a, {3,12,6}*864a
   5-fold covers : {3,6,30}*1080a, {15,6,6}*1080a
   6-fold covers : {9,6,12}*1296a, {3,6,36}*1296a, {3,6,12}*1296a, {3,6,12}*1296b, {6,6,18}*1296a, {18,6,6}*1296a, {6,6,6}*1296a, {6,6,6}*1296b, {3,6,12}*1296c, {6,6,6}*1296n, {6,6,6}*1296p
   7-fold covers : {3,6,42}*1512a, {21,6,6}*1512a
   8-fold covers : {3,6,48}*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, {3,12,12}*1728a, {3,24,6}*1728a, {6,12,6}*1728a, {6,12,6}*1728b
   9-fold covers : {9,6,18}*1944a, {3,6,6}*1944a, {3,6,54}*1944a, {27,6,6}*1944a, {3,6,18}*1944a, {9,6,6}*1944a, {3,6,18}*1944b, {9,6,6}*1944b, {3,6,18}*1944d, {9,6,6}*1944d, {3,6,6}*1944c, {3,6,6}*1944d, {3,6,6}*1944e, {3,6,6}*1944f
Permutation Representation (GAP) :
s0 := ( 2, 3)( 5, 6)( 8, 9)(11,12)(14,15)(17,18);;
s1 := ( 2, 3)( 4, 5)( 7, 9)(11,12)(13,14)(16,18);;
s2 := ( 1, 4)( 2, 6)( 3, 5)( 8, 9)(10,13)(11,15)(12,14)(17,18);;
s3 := ( 1,10)( 2,12)( 3,11)( 4,16)( 5,18)( 6,17)( 7,13)( 8,15)( 9,14);;
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, 
s0*s3*s0*s3, s1*s3*s1*s3, s0*s1*s0*s1*s0*s1, 
s2*s0*s1*s2*s1*s2*s0*s1*s2*s1, s3*s1*s2*s1*s2*s3*s1*s2*s1*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(18)!( 2, 3)( 5, 6)( 8, 9)(11,12)(14,15)(17,18);
s1 := Sym(18)!( 2, 3)( 4, 5)( 7, 9)(11,12)(13,14)(16,18);
s2 := Sym(18)!( 1, 4)( 2, 6)( 3, 5)( 8, 9)(10,13)(11,15)(12,14)(17,18);
s3 := Sym(18)!( 1,10)( 2,12)( 3,11)( 4,16)( 5,18)( 6,17)( 7,13)( 8,15)( 9,14);
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, 
s0*s1*s0*s1*s0*s1, s2*s0*s1*s2*s1*s2*s0*s1*s2*s1, 
s3*s1*s2*s1*s2*s3*s1*s2*s1*s2, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >; 
 
References : None.
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