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Polytope of Type {2,6,3}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,6,3}*216
if this polytope has a name.
Group : SmallGroup(216,102)
Rank : 4
Schlafli Type : {2,6,3}
Number of vertices, edges, etc : 2, 18, 27, 9
Order of s0s1s2s3 : 6
Order of s0s1s2s3s2s1 : 2
Special Properties :
   Degenerate
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {2,6,3,2} of size 432
   {2,6,3,4} of size 864
   {2,6,3,6} of size 1296
   {2,6,3,4} of size 1728
Vertex Figure Of :
   {2,2,6,3} of size 432
   {3,2,6,3} of size 648
   {4,2,6,3} of size 864
   {5,2,6,3} of size 1080
   {6,2,6,3} of size 1296
   {7,2,6,3} of size 1512
   {8,2,6,3} of size 1728
   {9,2,6,3} of size 1944
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {2,6,3}*72
   9-fold quotients : {2,2,3}*24
Covers (Minimal Covers in Boldface) :
   2-fold covers : {4,6,3}*432a, {2,6,6}*432a
   3-fold covers : {2,6,9}*648a, {2,6,9}*648b, {2,6,9}*648c, {2,6,9}*648d, {2,6,3}*648, {2,18,3}*648, {6,6,3}*648c
   4-fold covers : {8,6,3}*864a, {2,6,12}*864a, {4,6,6}*864c, {2,12,6}*864c, {2,6,3}*864, {2,12,3}*864
   5-fold covers : {10,6,3}*1080, {2,6,15}*1080
   6-fold covers : {4,6,9}*1296a, {4,6,9}*1296b, {4,6,9}*1296c, {4,6,9}*1296d, {4,6,3}*1296a, {4,18,3}*1296, {2,6,18}*1296a, {2,6,18}*1296c, {2,6,18}*1296d, {2,6,18}*1296e, {2,6,6}*1296c, {2,18,6}*1296h, {12,6,3}*1296e, {6,6,6}*1296f, {2,6,6}*1296f
   7-fold covers : {14,6,3}*1512, {2,6,21}*1512
   8-fold covers : {16,6,3}*1728a, {2,6,24}*1728a, {2,12,12}*1728b, {4,6,12}*1728b, {8,6,6}*1728c, {2,24,6}*1728c, {4,12,6}*1728c, {2,12,3}*1728, {2,24,3}*1728, {4,6,3}*1728a, {4,12,3}*1728a, {2,6,6}*1728b, {2,12,6}*1728a
   9-fold covers : {2,18,9}*1944a, {2,6,9}*1944a, {2,18,3}*1944a, {2,6,9}*1944b, {2,18,9}*1944b, {2,6,9}*1944c, {2,18,9}*1944c, {2,18,9}*1944d, {2,18,9}*1944e, {2,6,27}*1944a, {2,6,9}*1944d, {2,18,9}*1944f, {2,18,9}*1944g, {2,18,9}*1944h, {2,18,9}*1944i, {2,6,9}*1944e, {2,18,9}*1944j, {2,6,27}*1944b, {2,6,27}*1944c, {2,6,3}*1944, {2,18,3}*1944b, {6,6,9}*1944c, {18,6,3}*1944e, {6,6,3}*1944b, {6,6,3}*1944c, {6,6,9}*1944f, {6,6,9}*1944g, {6,6,9}*1944h, {6,6,3}*1944h, {6,18,3}*1944
Permutation Representation (GAP) :
s0 := (1,2);;
s1 := ( 6, 7)( 8, 9)(10,11);;
s2 := (4,8)(5,6)(7,9);;
s3 := ( 3, 4)( 6,10)( 7,11);;
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*s1*s0*s1, 
s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s2*s3*s2*s3*s2*s3, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s1*s3*s2*s1*s3*s2*s1*s3*s2*s1*s3*s2*s1*s3*s2*s1*s3*s2 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(11)!(1,2);
s1 := Sym(11)!( 6, 7)( 8, 9)(10,11);
s2 := Sym(11)!(4,8)(5,6)(7,9);
s3 := Sym(11)!( 3, 4)( 6,10)( 7,11);
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*s1*s0*s1, s0*s2*s0*s2, s0*s3*s0*s3, 
s1*s3*s1*s3, s2*s3*s2*s3*s2*s3, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s1*s3*s2*s1*s3*s2*s1*s3*s2*s1*s3*s2*s1*s3*s2*s1*s3*s2 >; 
 

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