Polytope of Type {2,6,6}

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

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

Permutation Representation (Magma) :

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

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