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Polytope of Type {18,4,2,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {18,4,2,6}*1728a
if this polytope has a name.
Group : SmallGroup(1728,30790)
Rank : 5
Schlafli Type : {18,4,2,6}
Number of vertices, edges, etc : 18, 36, 4, 6, 6
Order of s0s1s2s3s4 : 36
Order of s0s1s2s3s4s3s2s1 : 2
Special Properties :
   Degenerate
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {18,4,2,3}*864a, {18,2,2,6}*864
   3-fold quotients : {18,4,2,2}*576a, {6,4,2,6}*576a
   4-fold quotients : {9,2,2,6}*432, {18,2,2,3}*432
   6-fold quotients : {18,2,2,2}*288, {6,4,2,3}*288a, {6,2,2,6}*288
   8-fold quotients : {9,2,2,3}*216
   9-fold quotients : {2,4,2,6}*192, {6,4,2,2}*192a
   12-fold quotients : {9,2,2,2}*144, {3,2,2,6}*144, {6,2,2,3}*144
   18-fold quotients : {2,4,2,3}*96, {2,2,2,6}*96, {6,2,2,2}*96
   24-fold quotients : {3,2,2,3}*72
   27-fold quotients : {2,4,2,2}*64
   36-fold quotients : {2,2,2,3}*48, {3,2,2,2}*48
   54-fold quotients : {2,2,2,2}*32
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :
s0 := ( 2, 3)( 4, 8)( 5, 7)( 6, 9)(11,12)(13,17)(14,16)(15,18)(20,21)(22,26)
(23,25)(24,27)(29,30)(31,35)(32,34)(33,36);;
s1 := ( 1, 4)( 2, 6)( 3, 5)( 7, 8)(10,13)(11,15)(12,14)(16,17)(19,31)(20,33)
(21,32)(22,28)(23,30)(24,29)(25,35)(26,34)(27,36);;
s2 := ( 1,19)( 2,20)( 3,21)( 4,22)( 5,23)( 6,24)( 7,25)( 8,26)( 9,27)(10,28)
(11,29)(12,30)(13,31)(14,32)(15,33)(16,34)(17,35)(18,36);;
s3 := (39,40)(41,42);;
s4 := (37,41)(38,39)(40,42);;
poly := Group([s0,s1,s2,s3,s4]);;
 
Finitely Presented Group Representation (GAP) :
F := FreeGroup("s0","s1","s2","s3","s4");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  s4 := F.5;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s4*s4, s0*s2*s0*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s2*s3*s2*s3, 
s0*s4*s0*s4, s1*s4*s1*s4, s2*s4*s2*s4, 
s0*s1*s2*s1*s0*s1*s2*s1, s1*s2*s1*s2*s1*s2*s1*s2, 
s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*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(42)!( 2, 3)( 4, 8)( 5, 7)( 6, 9)(11,12)(13,17)(14,16)(15,18)(20,21)
(22,26)(23,25)(24,27)(29,30)(31,35)(32,34)(33,36);
s1 := Sym(42)!( 1, 4)( 2, 6)( 3, 5)( 7, 8)(10,13)(11,15)(12,14)(16,17)(19,31)
(20,33)(21,32)(22,28)(23,30)(24,29)(25,35)(26,34)(27,36);
s2 := Sym(42)!( 1,19)( 2,20)( 3,21)( 4,22)( 5,23)( 6,24)( 7,25)( 8,26)( 9,27)
(10,28)(11,29)(12,30)(13,31)(14,32)(15,33)(16,34)(17,35)(18,36);
s3 := Sym(42)!(39,40)(41,42);
s4 := Sym(42)!(37,41)(38,39)(40,42);
poly := sub<Sym(42)|s0,s1,s2,s3,s4>;
 
Finitely Presented Group Representation (Magma) :
poly<s0,s1,s2,s3,s4> := Group< s0,s1,s2,s3,s4 | s0*s0, s1*s1, s2*s2, 
s3*s3, s4*s4, s0*s2*s0*s2, s0*s3*s0*s3, 
s1*s3*s1*s3, s2*s3*s2*s3, s0*s4*s0*s4, 
s1*s4*s1*s4, s2*s4*s2*s4, s0*s1*s2*s1*s0*s1*s2*s1, 
s1*s2*s1*s2*s1*s2*s1*s2, s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*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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