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Polytope of Type {6,3,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,3,6}*216
Also Known As : 6T4(1,1)(1,1)if this polytope has another name.
Group : SmallGroup(216,162)
Rank : 4
Schlafli Type : {6,3,6}
Number of vertices, edges, etc : 6, 9, 9, 6
Order of s0s1s2s3 : 6
Order of s0s1s2s3s2s1 : 2
Special Properties :
   Universal
   Locally Toroidal
   Orientable
   Flat
   Self-Dual
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {6,3,6,2} of size 432
   {6,3,6,3} of size 648
   {6,3,6,4} of size 864
   {6,3,6,6} of size 1296
   {6,3,6,6} of size 1296
   {6,3,6,8} of size 1728
   {6,3,6,9} of size 1944
   {6,3,6,3} of size 1944
Vertex Figure Of :
   {2,6,3,6} of size 432
   {3,6,3,6} of size 648
   {4,6,3,6} of size 864
   {6,6,3,6} of size 1296
   {6,6,3,6} of size 1296
   {8,6,3,6} of size 1728
   {9,6,3,6} of size 1944
   {3,6,3,6} of size 1944
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {2,3,6}*72, {6,3,2}*72
   9-fold quotients : {2,3,2}*24
Covers (Minimal Covers in Boldface) :
   2-fold covers : {6,6,6}*432f
   3-fold covers : {6,9,6}*648, {6,3,6}*648a, {6,3,6}*648b
   4-fold covers : {6,12,6}*864e, {6,6,12}*864g, {12,6,6}*864g, {6,3,6}*864a, {6,3,6}*864b, {6,3,12}*864, {12,3,6}*864
   5-fold covers : {6,15,6}*1080
   6-fold covers : {6,18,6}*1296d, {6,6,6}*1296e, {6,6,6}*1296l, {6,6,6}*1296r, {6,6,6}*1296t
   7-fold covers : {6,21,6}*1512
   8-fold covers : {6,24,6}*1728e, {6,12,12}*1728f, {12,12,6}*1728e, {6,6,24}*1728g, {24,6,6}*1728g, {12,6,12}*1728g, {6,3,12}*1728, {6,3,24}*1728, {12,3,6}*1728, {24,3,6}*1728, {6,6,6}*1728c, {6,6,6}*1728e, {6,6,12}*1728d, {12,6,6}*1728d
   9-fold covers : {6,9,18}*1944, {18,9,6}*1944, {6,9,6}*1944a, {6,9,6}*1944b, {6,3,6}*1944a, {6,3,6}*1944b, {6,3,6}*1944c, {6,27,6}*1944, {6,9,6}*1944c, {6,9,6}*1944d, {6,9,6}*1944e, {6,9,6}*1944f, {6,9,6}*1944g, {6,9,6}*1944h, {6,3,6}*1944d, {6,3,6}*1944e, {6,3,18}*1944, {18,3,6}*1944
Permutation Representation (GAP) :
s0 := (10,19)(11,20)(12,21)(13,22)(14,23)(15,24)(16,25)(17,26)(18,27);;
s1 := ( 1,10)( 2,12)( 3,11)( 4,16)( 5,18)( 6,17)( 7,13)( 8,15)( 9,14)(20,21)
(22,25)(23,27)(24,26);;
s2 := ( 1, 5)( 2, 4)( 3, 6)( 7, 8)(10,23)(11,22)(12,24)(13,20)(14,19)(15,21)
(16,26)(17,25)(18,27);;
s3 := ( 4, 7)( 5, 8)( 6, 9)(13,16)(14,17)(15,18)(22,25)(23,26)(24,27);;
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, s1*s2*s1*s2*s1*s2, 
s2*s0*s1*s0*s1*s2*s0*s1*s0*s1, s3*s1*s2*s3*s2*s3*s1*s2*s3*s2 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(27)!(10,19)(11,20)(12,21)(13,22)(14,23)(15,24)(16,25)(17,26)(18,27);
s1 := Sym(27)!( 1,10)( 2,12)( 3,11)( 4,16)( 5,18)( 6,17)( 7,13)( 8,15)( 9,14)
(20,21)(22,25)(23,27)(24,26);
s2 := Sym(27)!( 1, 5)( 2, 4)( 3, 6)( 7, 8)(10,23)(11,22)(12,24)(13,20)(14,19)
(15,21)(16,26)(17,25)(18,27);
s3 := Sym(27)!( 4, 7)( 5, 8)( 6, 9)(13,16)(14,17)(15,18)(22,25)(23,26)(24,27);
poly := sub<Sym(27)|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, 
s1*s2*s1*s2*s1*s2, s2*s0*s1*s0*s1*s2*s0*s1*s0*s1, 
s3*s1*s2*s3*s2*s3*s1*s2*s3*s2 >; 
 
References :
  1. Theorem 11C7,11C8, McMullen P., Schulte, E.; Abstract Regular Polytopes (\ Cambridge University Press, 2002)

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