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

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,6,2,3}*432b
if this polytope has a name.
Group : SmallGroup(432,759)
Rank : 5
Schlafli Type : {6,6,2,3}
Number of vertices, edges, etc : 6, 18, 6, 3, 3
Order of s₀s₁s₂s₃s₄ : 6
Order of s₀s₁s₂s₃s₄s₃s₂s₁ : 2
Special Properties :
   Degenerate
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {6,6,2,3,2} of size 864
   {6,6,2,3,3} of size 1728
   {6,6,2,3,4} of size 1728
Vertex Figure Of :
   {2,6,6,2,3} of size 864
   {3,6,6,2,3} of size 1296
   {4,6,6,2,3} of size 1728
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {6,3,2,3}*216
   3-fold quotients : {2,6,2,3}*144
   6-fold quotients : {2,3,2,3}*72
   9-fold quotients : {2,2,2,3}*48
Covers (Minimal Covers in Boldface) :
   2-fold covers : {6,12,2,3}*864b, {12,6,2,3}*864c, {6,6,2,6}*864b
   3-fold covers : {6,6,2,9}*1296b, {6,18,2,3}*1296b, {6,6,6,3}*1296b, {6,6,2,3}*1296a, {6,6,6,3}*1296d, {6,6,2,3}*1296d
   4-fold covers : {6,24,2,3}*1728b, {12,12,2,3}*1728b, {24,6,2,3}*1728c, {6,6,2,12}*1728b, {6,12,2,6}*1728b, {6,6,4,6}*1728b, {12,6,2,6}*1728c, {6,6,4,3}*1728b, {6,6,2,3}*1728a, {12,6,2,3}*1728b
Permutation Representation (GAP) :

s0 := ( 5, 6)( 9,10)(11,12)(13,14)(15,16)(17,18);;
s1 := ( 1, 5)( 2, 9)( 3,13)( 4,11)( 7,17)( 8,15)(12,14)(16,18);;
s2 := ( 1, 7)( 2, 3)( 4, 8)( 5,16)( 6,15)( 9,12)(10,11)(13,18)(14,17);;
s3 := (20,21);;
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, s2*s3*s2*s3, 
s0*s4*s0*s4, s1*s4*s1*s4, s2*s4*s2*s4, 
s3*s4*s3*s4*s3*s4, s2*s0*s1*s0*s1*s2*s0*s1*s0*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;

Permutation Representation (Magma) :

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

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