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

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

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