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

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,2,6,6}*1728d
if this polytope has a name.
Group : SmallGroup(1728,47341)
Rank : 5
Schlafli Type : {4,2,6,6}
Number of vertices, edges, etc : 4, 4, 18, 54, 18
Order of s₀s₁s₂s₃s₄ : 12
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 : {2,2,6,6}*864d
   3-fold quotients : {4,2,6,6}*576a, {4,2,6,6}*576b, {4,2,6,6}*576c
   6-fold quotients : {4,2,3,6}*288, {4,2,6,3}*288, {2,2,6,6}*288a, {2,2,6,6}*288b, {2,2,6,6}*288c
   9-fold quotients : {4,2,2,6}*192, {4,2,6,2}*192
   12-fold quotients : {2,2,3,6}*144, {2,2,6,3}*144
   18-fold quotients : {4,2,2,3}*96, {4,2,3,2}*96, {2,2,2,6}*96, {2,2,6,2}*96
   27-fold quotients : {4,2,2,2}*64
   36-fold quotients : {2,2,2,3}*48, {2,2,3,2}*48
   54-fold quotients : {2,2,2,2}*32
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := (2,3);;
s1 := (1,2)(3,4);;
s2 := ( 8,11)( 9,12)(10,13)(14,23)(15,24)(16,25)(17,29)(18,30)(19,31)(20,26)
(21,27)(22,28);;
s3 := ( 5,17)( 6,19)( 7,18)( 8,14)( 9,16)(10,15)(11,20)(12,22)(13,21)(23,26)
(24,28)(25,27)(30,31);;
s4 := ( 5, 6)( 8, 9)(11,12)(14,24)(15,23)(16,25)(17,27)(18,26)(19,28)(20,30)
(21,29)(22,31);;
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, 
s0*s1*s0*s1*s0*s1*s0*s1, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4, 
s4*s2*s3*s4*s3*s2*s3*s4*s2*s3*s4*s3*s2*s3 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(31)!(2,3);
s1 := Sym(31)!(1,2)(3,4);
s2 := Sym(31)!( 8,11)( 9,12)(10,13)(14,23)(15,24)(16,25)(17,29)(18,30)(19,31)
(20,26)(21,27)(22,28);
s3 := Sym(31)!( 5,17)( 6,19)( 7,18)( 8,14)( 9,16)(10,15)(11,20)(12,22)(13,21)
(23,26)(24,28)(25,27)(30,31);
s4 := Sym(31)!( 5, 6)( 8, 9)(11,12)(14,24)(15,23)(16,25)(17,27)(18,26)(19,28)
(20,30)(21,29)(22,31);
poly := sub<Sym(31)|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, s0*s1*s0*s1*s0*s1*s0*s1, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4, 
s4*s2*s3*s4*s3*s2*s3*s4*s2*s3*s4*s3*s2*s3 >;

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