Polytope of Type {6,6,2}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,6,2}*432b
if this polytope has a name.
Group : SmallGroup(432,545)
Rank : 4
Schlafli Type : {6,6,2}
Number of vertices, edges, etc : 18, 54, 18, 2
Order of s₀s₁s₂s₃ : 6
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Degenerate
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {6,6,2,2} of size 864
   {6,6,2,3} of size 1296
   {6,6,2,4} of size 1728
Vertex Figure Of :
   {2,6,6,2} of size 864
   {3,6,6,2} of size 1296
   {4,6,6,2} of size 1728
   {4,6,6,2} of size 1728
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {6,6,2}*216
   3-fold quotients : {6,6,2}*144a
   9-fold quotients : {2,6,2}*48, {6,2,2}*48
   18-fold quotients : {2,3,2}*24, {3,2,2}*24
   27-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   2-fold covers : {6,12,2}*864b, {12,6,2}*864b, {6,6,4}*864b
   3-fold covers : {6,18,2}*1296b, {18,6,2}*1296b, {6,6,2}*1296a, {6,6,2}*1296b, {6,18,2}*1296f, {18,6,2}*1296f, {6,18,2}*1296g, {18,6,2}*1296g, {6,6,6}*1296j, {6,6,6}*1296m, {6,6,2}*1296g
   4-fold covers : {12,6,4}*1728a, {6,12,4}*1728b, {6,24,2}*1728b, {24,6,2}*1728b, {6,6,8}*1728b, {12,12,2}*1728c, {6,6,4}*1728b, {6,12,2}*1728b, {12,6,2}*1728b
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

to this polytope