Polytope of Type {6,4,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,4,6}*432a
if this polytope has a name.
Group : SmallGroup(432,741)
Rank : 4
Schlafli Type : {6,4,6}
Number of vertices, edges, etc : 9, 18, 18, 6
Order of s₀s₁s₂s₃ : 12
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Universal
   Non-Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {6,4,6,2} of size 864
   {6,4,6,3} of size 1296
   {6,4,6,4} of size 1728
   {6,4,6,3} of size 1728
   {6,4,6,4} of size 1728
Vertex Figure Of :
   {2,6,4,6} of size 864
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {6,4,2}*144
Covers (Minimal Covers in Boldface) :
   2-fold covers : {6,4,12}*864, {6,4,6}*864b
   3-fold covers : {6,4,18}*1296, {6,4,6}*1296a, {6,12,6}*1296a, {6,12,6}*1296b, {6,12,6}*1296e, {6,12,6}*1296g
   4-fold covers : {6,4,24}*1728, {6,8,6}*1728a, {6,4,12}*1728a, {12,4,6}*1728b
Permutation Representation (GAP) :

s0 := ( 4, 7)( 5, 8)( 6, 9)(13,16)(14,17)(15,18);;
s1 := ( 1,13)( 2,14)( 3,15)( 4,16)( 5,17)( 6,18)( 7,10)( 8,11)( 9,12);;
s2 := ( 2, 3)( 4, 7)( 5, 9)( 6, 8)(11,12)(14,15)(17,18);;
s3 := ( 1, 2)( 4, 5)( 7, 8)(10,11)(13,14)(16,17);;
poly := Group([s0,s1,s2,s3]);;

Finitely Presented Group Representation (GAP) :

F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s2*s0*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s1*s2*s1*s2*s1*s2*s1*s2, 
s1*s2*s3*s2*s1*s2*s3*s2, s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s0*s1*s2*s1*s0*s1*s0*s1*s2*s0*s1*s0*s1 ];;
poly := F / rels;;

Permutation Representation (Magma) :

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

Finitely Presented Group Representation (Magma) :

poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2, 
s3*s3, s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s1*s2*s1*s2*s1*s2*s1*s2, s1*s2*s3*s2*s1*s2*s3*s2, 
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s0*s1*s2*s1*s0*s1*s0*s1*s2*s0*s1*s0*s1 >;

References : None.
to this polytope