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Polytope of Type {9,4,2}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {9,4,2}*144
if this polytope has a name.
Group : SmallGroup(144,109)
Rank : 4
Schlafli Type : {9,4,2}
Number of vertices, edges, etc : 9, 18, 4, 2
Order of s0s1s2s3 : 18
Order of s0s1s2s3s2s1 : 2
Special Properties :
   Degenerate
   Universal
   Non-Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {9,4,2,2} of size 288
   {9,4,2,3} of size 432
   {9,4,2,4} of size 576
   {9,4,2,5} of size 720
   {9,4,2,6} of size 864
   {9,4,2,7} of size 1008
   {9,4,2,8} of size 1152
   {9,4,2,9} of size 1296
   {9,4,2,10} of size 1440
   {9,4,2,11} of size 1584
   {9,4,2,12} of size 1728
   {9,4,2,13} of size 1872
Vertex Figure Of :
   {2,9,4,2} of size 288
   {4,9,4,2} of size 576
   {6,9,4,2} of size 864
   {4,9,4,2} of size 1152
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {3,4,2}*48
Covers (Minimal Covers in Boldface) :
   2-fold covers : {9,4,2}*288, {18,4,2}*288b, {18,4,2}*288c
   3-fold covers : {27,4,2}*432
   4-fold covers : {9,4,4}*576a, {36,4,2}*576b, {36,4,2}*576c, {9,4,4}*576b, {9,8,2}*576, {18,4,2}*576
   5-fold covers : {45,4,2}*720
   6-fold covers : {27,4,2}*864, {54,4,2}*864b, {54,4,2}*864c, {9,4,6}*864, {9,12,2}*864, {18,12,2}*864c
   7-fold covers : {63,4,2}*1008
   8-fold covers : {9,4,4}*1152a, {9,4,4}*1152b, {18,4,4}*1152b, {18,4,4}*1152c, {18,4,2}*1152a, {9,8,2}*1152, {18,8,2}*1152a, {9,8,4}*1152, {72,4,2}*1152c, {72,4,2}*1152d, {9,4,8}*1152, {36,4,2}*1152b, {18,4,4}*1152d, {18,4,2}*1152b, {36,4,2}*1152c, {18,8,2}*1152b, {18,8,2}*1152c
   9-fold covers : {81,4,2}*1296
   10-fold covers : {9,4,10}*1440, {18,20,2}*1440b, {45,4,2}*1440, {90,4,2}*1440b, {90,4,2}*1440c
   11-fold covers : {99,4,2}*1584
   12-fold covers : {27,4,4}*1728a, {108,4,2}*1728b, {108,4,2}*1728c, {27,4,4}*1728b, {27,8,2}*1728, {54,4,2}*1728, {9,4,12}*1728, {9,24,2}*1728, {9,8,6}*1728, {9,12,4}*1728, {18,4,6}*1728b, {18,12,2}*1728a, {18,12,2}*1728b
   13-fold covers : {117,4,2}*1872
Permutation Representation (GAP) :
s0 := ( 1, 2)( 3, 6)( 4, 5)( 7,15)( 8,14)( 9,16)(10,12)(11,13)(17,23)(18,24)
(19,21)(20,22)(25,31)(26,32)(27,29)(28,30)(33,36)(34,35);;
s1 := ( 1, 5)( 2, 3)( 4,12)( 6, 8)( 7, 9)(10,21)(11,22)(13,15)(14,17)(16,18)
(19,29)(20,30)(23,25)(24,26)(27,31)(28,35)(32,33)(34,36);;
s2 := ( 1,15)( 2, 7)( 3, 9)( 6,16)(10,20)(12,22)(17,26)(19,28)(21,30)(23,32)
(25,33)(31,36);;
s3 := (37,38);;
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, 
s1*s2*s1*s2*s1*s2*s1*s2, s2*s1*s0*s2*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(38)!( 1, 2)( 3, 6)( 4, 5)( 7,15)( 8,14)( 9,16)(10,12)(11,13)(17,23)
(18,24)(19,21)(20,22)(25,31)(26,32)(27,29)(28,30)(33,36)(34,35);
s1 := Sym(38)!( 1, 5)( 2, 3)( 4,12)( 6, 8)( 7, 9)(10,21)(11,22)(13,15)(14,17)
(16,18)(19,29)(20,30)(23,25)(24,26)(27,31)(28,35)(32,33)(34,36);
s2 := Sym(38)!( 1,15)( 2, 7)( 3, 9)( 6,16)(10,20)(12,22)(17,26)(19,28)(21,30)
(23,32)(25,33)(31,36);
s3 := Sym(38)!(37,38);
poly := sub<Sym(38)|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, s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s1*s0*s2*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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