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

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,2,18,4}*1728a
if this polytope has a name.
Group : SmallGroup(1728,30790)
Rank : 5
Schlafli Type : {6,2,18,4}
Number of vertices, edges, etc : 6, 6, 18, 36, 4
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 : {3,2,18,4}*864a, {6,2,18,2}*864
   3-fold quotients : {2,2,18,4}*576a, {6,2,6,4}*576a
   4-fold quotients : {3,2,18,2}*432, {6,2,9,2}*432
   6-fold quotients : {2,2,18,2}*288, {3,2,6,4}*288a, {6,2,6,2}*288
   8-fold quotients : {3,2,9,2}*216
   9-fold quotients : {2,2,6,4}*192a, {6,2,2,4}*192
   12-fold quotients : {2,2,9,2}*144, {3,2,6,2}*144, {6,2,3,2}*144
   18-fold quotients : {3,2,2,4}*96, {2,2,6,2}*96, {6,2,2,2}*96
   24-fold quotients : {3,2,3,2}*72
   27-fold quotients : {2,2,2,4}*64
   36-fold quotients : {2,2,3,2}*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 := (3,4)(5,6);;
s1 := (1,5)(2,3)(4,6);;
s2 := ( 8, 9)(10,14)(11,13)(12,15)(17,18)(19,23)(20,22)(21,24)(26,27)(28,32)
(29,31)(30,33)(35,36)(37,41)(38,40)(39,42);;
s3 := ( 7,10)( 8,12)( 9,11)(13,14)(16,19)(17,21)(18,20)(22,23)(25,37)(26,39)
(27,38)(28,34)(29,36)(30,35)(31,41)(32,40)(33,42);;
s4 := ( 7,25)( 8,26)( 9,27)(10,28)(11,29)(12,30)(13,31)(14,32)(15,33)(16,34)
(17,35)(18,36)(19,37)(20,38)(21,39)(22,40)(23,41)(24,42);;
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, 
s1*s2*s1*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s0*s4*s0*s4, s1*s4*s1*s4, s2*s4*s2*s4, 
s2*s3*s4*s3*s2*s3*s4*s3, s3*s4*s3*s4*s3*s4*s3*s4, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*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(42)!(3,4)(5,6);
s1 := Sym(42)!(1,5)(2,3)(4,6);
s2 := Sym(42)!( 8, 9)(10,14)(11,13)(12,15)(17,18)(19,23)(20,22)(21,24)(26,27)
(28,32)(29,31)(30,33)(35,36)(37,41)(38,40)(39,42);
s3 := Sym(42)!( 7,10)( 8,12)( 9,11)(13,14)(16,19)(17,21)(18,20)(22,23)(25,37)
(26,39)(27,38)(28,34)(29,36)(30,35)(31,41)(32,40)(33,42);
s4 := Sym(42)!( 7,25)( 8,26)( 9,27)(10,28)(11,29)(12,30)(13,31)(14,32)(15,33)
(16,34)(17,35)(18,36)(19,37)(20,38)(21,39)(22,40)(23,41)(24,42);
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, s1*s2*s1*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s0*s4*s0*s4, 
s1*s4*s1*s4, s2*s4*s2*s4, s2*s3*s4*s3*s2*s3*s4*s3, 
s3*s4*s3*s4*s3*s4*s3*s4, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >;

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