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Polytope of Type {2,6,9}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,6,9}*216
if this polytope has a name.
Group : SmallGroup(216,101)
Rank : 4
Schlafli Type : {2,6,9}
Number of vertices, edges, etc : 2, 6, 27, 9
Order of s0s1s2s3 : 18
Order of s0s1s2s3s2s1 : 2
Special Properties :
   Degenerate
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {2,6,9,2} of size 432
   {2,6,9,4} of size 864
   {2,6,9,6} of size 1296
   {2,6,9,4} of size 1728
Vertex Figure Of :
   {2,2,6,9} of size 432
   {3,2,6,9} of size 648
   {4,2,6,9} of size 864
   {5,2,6,9} of size 1080
   {6,2,6,9} of size 1296
   {7,2,6,9} of size 1512
   {8,2,6,9} of size 1728
   {9,2,6,9} of size 1944
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {2,2,9}*72, {2,6,3}*72
   9-fold quotients : {2,2,3}*24
Covers (Minimal Covers in Boldface) :
   2-fold covers : {4,6,9}*432, {2,6,18}*432b
   3-fold covers : {2,18,9}*648, {2,6,9}*648a, {2,6,27}*648, {6,6,9}*648b
   4-fold covers : {8,6,9}*864, {2,6,36}*864b, {4,6,18}*864b, {2,12,18}*864b, {2,6,9}*864, {2,12,9}*864
   5-fold covers : {10,6,9}*1080, {2,6,45}*1080
   6-fold covers : {4,18,9}*1296, {4,6,9}*1296a, {4,6,27}*1296, {2,18,18}*1296b, {2,6,18}*1296a, {2,6,54}*1296b, {12,6,9}*1296b, {6,6,18}*1296c, {2,6,18}*1296i
   7-fold covers : {14,6,9}*1512, {2,6,63}*1512
   8-fold covers : {16,6,9}*1728, {2,6,72}*1728b, {2,12,36}*1728b, {4,6,36}*1728b, {8,6,18}*1728b, {2,24,18}*1728b, {4,12,18}*1728b, {2,12,9}*1728, {4,6,9}*1728a, {2,24,9}*1728, {4,12,9}*1728, {2,6,18}*1728, {2,12,18}*1728b
   9-fold covers : {2,18,9}*1944a, {2,18,27}*1944, {2,6,27}*1944a, {2,6,9}*1944d, {2,18,9}*1944h, {2,18,9}*1944i, {2,6,9}*1944e, {2,6,27}*1944b, {2,6,27}*1944c, {2,6,81}*1944, {6,18,9}*1944, {18,6,9}*1944b, {6,6,9}*1944c, {6,6,9}*1944d, {6,6,9}*1944e, {6,6,27}*1944b
Permutation Representation (GAP) :
s0 := (1,2);;
s1 := ( 6, 7)( 9,10)(12,13)(15,16)(18,19)(21,22)(24,25)(26,27)(28,29);;
s2 := ( 3, 6)( 4,12)( 5, 9)( 8,18)(10,13)(11,15)(14,24)(16,19)(17,21)(20,28)
(22,25)(23,26)(27,29);;
s3 := ( 3, 4)( 5, 8)( 6,10)( 7, 9)(11,14)(12,16)(13,15)(17,20)(18,22)(19,21)
(24,27)(25,26)(28,29);;
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, 
s3*s1*s2*s1*s2*s3*s1*s2*s1*s2, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(29)!(1,2);
s1 := Sym(29)!( 6, 7)( 9,10)(12,13)(15,16)(18,19)(21,22)(24,25)(26,27)(28,29);
s2 := Sym(29)!( 3, 6)( 4,12)( 5, 9)( 8,18)(10,13)(11,15)(14,24)(16,19)(17,21)
(20,28)(22,25)(23,26)(27,29);
s3 := Sym(29)!( 3, 4)( 5, 8)( 6,10)( 7, 9)(11,14)(12,16)(13,15)(17,20)(18,22)
(19,21)(24,27)(25,26)(28,29);
poly := sub<Sym(29)|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, s3*s1*s2*s1*s2*s3*s1*s2*s1*s2, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >; 
 

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