Polytope of Type {3,6,6,2}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,6,6,2}*432a
if this polytope has a name.
Group : SmallGroup(432,545)
Rank : 5
Schlafli Type : {3,6,6,2}
Number of vertices, edges, etc : 3, 9, 18, 6, 2
Order of s₀s₁s₂s₃s₄ : 6
Order of s₀s₁s₂s₃s₄s₃s₂s₁ : 2
Special Properties :
   Degenerate
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {3,6,6,2,2} of size 864
   {3,6,6,2,3} of size 1296
   {3,6,6,2,4} of size 1728
Vertex Figure Of :
   {2,3,6,6,2} of size 864
   {4,3,6,6,2} of size 1728
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {3,6,3,2}*216
   3-fold quotients : {3,2,6,2}*144
   6-fold quotients : {3,2,3,2}*72
   9-fold quotients : {3,2,2,2}*48
Covers (Minimal Covers in Boldface) :
   2-fold covers : {3,6,12,2}*864a, {3,6,6,4}*864a, {6,6,6,2}*864a
   3-fold covers : {3,6,18,2}*1296a, {9,6,6,2}*1296a, {3,6,6,2}*1296a, {3,6,6,2}*1296b, {3,6,6,6}*1296a, {3,6,6,6}*1296b, {3,6,6,2}*1296e
   4-fold covers : {3,6,12,4}*1728a, {3,6,24,2}*1728a, {3,6,6,8}*1728a, {6,6,12,2}*1728a, {12,6,6,2}*1728a, {6,6,6,4}*1728a, {6,12,6,2}*1728a, {3,6,6,4}*1728a, {3,12,6,2}*1728a
Permutation Representation (GAP) :

s0 := ( 2, 3)( 5, 6)( 8, 9)(11,12)(14,15)(17,18);;
s1 := ( 2, 3)( 4, 5)( 7, 9)(11,12)(13,14)(16,18);;
s2 := ( 1, 4)( 2, 6)( 3, 5)( 8, 9)(10,13)(11,15)(12,14)(17,18);;
s3 := ( 1,10)( 2,12)( 3,11)( 4,16)( 5,18)( 6,17)( 7,13)( 8,15)( 9,14);;
s4 := (19,20);;
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, s0*s4*s0*s4, 
s1*s4*s1*s4, s2*s4*s2*s4, s3*s4*s3*s4, 
s0*s1*s0*s1*s0*s1, s2*s0*s1*s2*s1*s2*s0*s1*s2*s1, 
s3*s1*s2*s1*s2*s3*s1*s2*s1*s2, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(20)!( 2, 3)( 5, 6)( 8, 9)(11,12)(14,15)(17,18);
s1 := Sym(20)!( 2, 3)( 4, 5)( 7, 9)(11,12)(13,14)(16,18);
s2 := Sym(20)!( 1, 4)( 2, 6)( 3, 5)( 8, 9)(10,13)(11,15)(12,14)(17,18);
s3 := Sym(20)!( 1,10)( 2,12)( 3,11)( 4,16)( 5,18)( 6,17)( 7,13)( 8,15)( 9,14);
s4 := Sym(20)!(19,20);
poly := sub<Sym(20)|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, s0*s4*s0*s4, s1*s4*s1*s4, 
s2*s4*s2*s4, s3*s4*s3*s4, s0*s1*s0*s1*s0*s1, 
s2*s0*s1*s2*s1*s2*s0*s1*s2*s1, s3*s1*s2*s1*s2*s3*s1*s2*s1*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >;

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