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

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

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