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Polytope of Type {3,6,2}

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

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