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Polytope of Type {6,3,2}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,3,2}*216
if this polytope has a name.
Group : SmallGroup(216,102)
Rank : 4
Schlafli Type : {6,3,2}
Number of vertices, edges, etc : 18, 27, 9, 2
Order of s0s1s2s3 : 6
Order of s0s1s2s3s2s1 : 2
Special Properties :
   Degenerate
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {6,3,2,2} of size 432
   {6,3,2,3} of size 648
   {6,3,2,4} of size 864
   {6,3,2,5} of size 1080
   {6,3,2,6} of size 1296
   {6,3,2,7} of size 1512
   {6,3,2,8} of size 1728
   {6,3,2,9} of size 1944
Vertex Figure Of :
   {2,6,3,2} of size 432
   {3,6,3,2} of size 648
   {4,6,3,2} of size 864
   {6,6,3,2} of size 1296
   {6,6,3,2} of size 1296
   {8,6,3,2} of size 1728
   {3,6,3,2} of size 1944
   {9,6,3,2} of size 1944
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {6,3,2}*72
   9-fold quotients : {2,3,2}*24
Covers (Minimal Covers in Boldface) :
   2-fold covers : {6,6,2}*432a
   3-fold covers : {6,9,2}*648a, {6,9,2}*648b, {6,9,2}*648c, {6,9,2}*648d, {6,3,2}*648, {18,3,2}*648, {6,3,6}*648b
   4-fold covers : {6,12,2}*864a, {6,6,4}*864a, {12,6,2}*864c, {6,3,4}*864, {6,3,2}*864, {12,3,2}*864
   5-fold covers : {6,15,2}*1080
   6-fold covers : {6,18,2}*1296a, {6,18,2}*1296c, {6,18,2}*1296d, {6,18,2}*1296e, {6,6,2}*1296c, {18,6,2}*1296h, {6,6,6}*1296i, {6,6,6}*1296l, {6,6,2}*1296f
   7-fold covers : {6,21,2}*1512
   8-fold covers : {6,12,4}*1728a, {6,24,2}*1728a, {6,6,8}*1728a, {12,12,2}*1728b, {24,6,2}*1728c, {12,6,4}*1728c, {12,3,2}*1728, {24,3,2}*1728, {6,3,8}*1728, {6,6,4}*1728a, {6,6,2}*1728b, {12,6,2}*1728a
   9-fold covers : {18,9,2}*1944a, {6,9,2}*1944a, {18,3,2}*1944a, {6,9,2}*1944b, {18,9,2}*1944b, {6,9,2}*1944c, {18,9,2}*1944c, {18,9,2}*1944d, {18,9,2}*1944e, {6,27,2}*1944a, {6,9,2}*1944d, {18,9,2}*1944f, {18,9,2}*1944g, {18,9,2}*1944h, {18,9,2}*1944i, {6,9,2}*1944e, {18,9,2}*1944j, {6,27,2}*1944b, {6,27,2}*1944c, {6,3,2}*1944, {18,3,2}*1944b, {6,9,6}*1944b, {6,3,6}*1944a, {6,3,6}*1944c, {6,9,6}*1944e, {6,9,6}*1944f, {6,9,6}*1944h, {6,3,6}*1944e, {18,3,6}*1944
Permutation Representation (GAP) :
s0 := (4,5)(6,7)(8,9);;
s1 := (2,6)(3,4)(5,7);;
s2 := (1,2)(4,8)(5,9);;
s3 := (10,11);;
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, 
s1*s2*s1*s2*s1*s2, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(11)!(4,5)(6,7)(8,9);
s1 := Sym(11)!(2,6)(3,4)(5,7);
s2 := Sym(11)!(1,2)(4,8)(5,9);
s3 := Sym(11)!(10,11);
poly := sub<Sym(11)|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, s1*s2*s1*s2*s1*s2, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1 >; 
 

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