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

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,2,6,3}*216
if this polytope has a name.
Group : SmallGroup(216,162)
Rank : 5
Schlafli Type : {3,2,6,3}
Number of vertices, edges, etc : 3, 3, 6, 9, 3
Order of s0s1s2s3s4 : 6
Order of s0s1s2s3s4s3s2s1 : 2
Special Properties :
   Degenerate
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {3,2,6,3,2} of size 432
   {3,2,6,3,4} of size 864
   {3,2,6,3,6} of size 1296
   {3,2,6,3,4} of size 1728
Vertex Figure Of :
   {2,3,2,6,3} of size 432
   {3,3,2,6,3} of size 864
   {4,3,2,6,3} of size 864
   {6,3,2,6,3} of size 1296
   {4,3,2,6,3} of size 1728
   {6,3,2,6,3} of size 1728
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {3,2,2,3}*72
Covers (Minimal Covers in Boldface) :
   2-fold covers : {3,2,6,6}*432b, {6,2,6,3}*432
   3-fold covers : {3,2,6,9}*648, {9,2,6,3}*648, {3,2,6,3}*648, {3,6,6,3}*648
   4-fold covers : {3,2,6,12}*864b, {12,2,6,3}*864, {3,2,12,6}*864c, {6,4,6,3}*864, {3,4,6,3}*864, {3,2,6,3}*864, {3,2,12,3}*864, {6,2,6,6}*864b
   5-fold covers : {3,2,6,15}*1080, {15,2,6,3}*1080
   6-fold covers : {3,2,6,18}*1296b, {6,2,6,9}*1296, {9,2,6,6}*1296b, {18,2,6,3}*1296, {3,2,6,6}*1296a, {6,2,6,3}*1296, {3,2,6,6}*1296d, {6,6,6,3}*1296c, {3,6,6,6}*1296e, {6,6,6,3}*1296e
   7-fold covers : {3,2,6,21}*1512, {21,2,6,3}*1512
   8-fold covers : {3,2,6,24}*1728b, {24,2,6,3}*1728, {3,2,12,12}*1728b, {12,4,6,3}*1728, {3,2,24,6}*1728c, {6,8,6,3}*1728, {3,2,12,3}*1728, {3,2,24,3}*1728, {3,8,6,3}*1728, {6,2,6,12}*1728b, {12,2,6,6}*1728b, {6,4,6,6}*1728c, {6,2,12,6}*1728c, {3,4,6,6}*1728b, {6,4,6,3}*1728b, {3,2,6,6}*1728a, {3,2,12,6}*1728b, {6,2,6,3}*1728, {6,2,12,3}*1728
   9-fold covers : {3,2,18,9}*1944, {9,2,6,9}*1944, {3,2,6,9}*1944a, {9,2,6,3}*1944, {3,2,6,27}*1944, {27,2,6,3}*1944, {3,2,6,9}*1944b, {3,2,6,9}*1944c, {3,2,6,9}*1944d, {3,2,6,3}*1944, {3,2,18,3}*1944, {3,6,6,9}*1944, {9,6,6,3}*1944, {3,6,6,3}*1944a, {3,6,6,3}*1944b, {3,6,6,3}*1944c, {3,6,6,3}*1944d
Permutation Representation (GAP) :
s0 := (2,3);;
s1 := (1,2);;
s2 := ( 7, 8)( 9,10)(11,12);;
s3 := ( 4, 7)( 5,11)( 6, 9)(10,12);;
s4 := ( 4, 5)( 7,10)( 8, 9)(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, s3*s4*s3*s4*s3*s4, 
s4*s2*s3*s2*s3*s4*s2*s3*s2*s3 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(12)!(2,3);
s1 := Sym(12)!(1,2);
s2 := Sym(12)!( 7, 8)( 9,10)(11,12);
s3 := Sym(12)!( 4, 7)( 5,11)( 6, 9)(10,12);
s4 := Sym(12)!( 4, 5)( 7,10)( 8, 9)(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, 
s3*s4*s3*s4*s3*s4, s4*s2*s3*s2*s3*s4*s2*s3*s2*s3 >; 
 

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