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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}*864
if this polytope has a name.
Group : SmallGroup(864,3998)
Rank : 4
Schlafli Type : {2,6,9}
Number of vertices, edges, etc : 2, 24, 108, 36
Order of s0s1s2s3 : 36
Order of s0s1s2s3s2s1 : 2
Special Properties :
   Degenerate
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {2,6,9,2} of size 1728
Vertex Figure Of :
   {2,2,6,9} of size 1728
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {2,6,3}*288
   4-fold quotients : {2,6,9}*216
   9-fold quotients : {2,6,3}*96
   12-fold quotients : {2,2,9}*72, {2,6,3}*72
   18-fold quotients : {2,3,3}*48
   36-fold quotients : {2,2,3}*24
Covers (Minimal Covers in Boldface) :
   2-fold covers : {2,12,9}*1728, {4,6,9}*1728a, {2,6,18}*1728
Permutation Representation (GAP) :
s0 := (1,2);;
s1 := ( 4, 5)( 8, 9)(12,13)(16,17)(20,21)(24,25)(28,29)(32,33)(36,37);;
s2 := ( 5, 6)( 7,11)( 8,12)( 9,14)(10,13)(15,31)(16,32)(17,34)(18,33)(19,27)
(20,28)(21,30)(22,29)(23,35)(24,36)(25,38)(26,37);;
s3 := ( 3,18)( 4,16)( 5,17)( 6,15)( 7,26)( 8,24)( 9,25)(10,23)(11,22)(12,20)
(13,21)(14,19)(27,34)(28,32)(29,33)(30,31)(35,38);;
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*s1*s0*s1, 
s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s1*s2*s3*s2*s3*s2*s1*s2*s3*s2*s3*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s3*s1*s2*s1*s2*s3*s1*s2*s1*s2*s3*s1*s2*s1*s2*s3*s1*s2*s1*s2 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(38)!(1,2);
s1 := Sym(38)!( 4, 5)( 8, 9)(12,13)(16,17)(20,21)(24,25)(28,29)(32,33)(36,37);
s2 := Sym(38)!( 5, 6)( 7,11)( 8,12)( 9,14)(10,13)(15,31)(16,32)(17,34)(18,33)
(19,27)(20,28)(21,30)(22,29)(23,35)(24,36)(25,38)(26,37);
s3 := Sym(38)!( 3,18)( 4,16)( 5,17)( 6,15)( 7,26)( 8,24)( 9,25)(10,23)(11,22)
(12,20)(13,21)(14,19)(27,34)(28,32)(29,33)(30,31)(35,38);
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*s1*s0*s1, s0*s2*s0*s2, s0*s3*s0*s3, 
s1*s3*s1*s3, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s1*s2*s3*s2*s3*s2*s1*s2*s3*s2*s3*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s3*s1*s2*s1*s2*s3*s1*s2*s1*s2*s3*s1*s2*s1*s2*s3*s1*s2*s1*s2 >; 
 

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