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Polytope of Type {9,6,2}

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

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