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Polytope of Type {2,6,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,6,4}*864b
if this polytope has a name.
Group : SmallGroup(864,4686)
Rank : 4
Schlafli Type : {2,6,4}
Number of vertices, edges, etc : 2, 54, 108, 36
Order of s0s1s2s3 : 12
Order of s0s1s2s3s2s1 : 2
Special Properties :
   Degenerate
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {2,6,4,2} of size 1728
Vertex Figure Of :
   {2,2,6,4} of size 1728
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {2,6,4}*288
   6-fold quotients : {2,6,4}*144
   9-fold quotients : {2,6,4}*96a
   18-fold quotients : {2,6,2}*48
   27-fold quotients : {2,2,4}*32
   36-fold quotients : {2,3,2}*24
   54-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   2-fold covers : {2,12,4}*1728d, {2,6,8}*1728b, {4,6,4}*1728d
Permutation Representation (GAP) :
s0 := (1,2);;
s1 := ( 4, 5)( 6, 9)( 7,11)( 8,10)(13,14)(15,18)(16,20)(17,19);;
s2 := ( 3,16)( 4,15)( 5,17)( 6,19)( 7,18)( 8,20)( 9,13)(10,12)(11,14);;
s3 := ( 6, 9)( 7,10)( 8,11);;
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, 
s2*s3*s2*s3*s2*s3*s2*s3, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s3*s1*s2*s1*s2*s3*s2*s1*s2*s3*s1*s2*s1*s2*s3*s2*s1*s2 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(20)!(1,2);
s1 := Sym(20)!( 4, 5)( 6, 9)( 7,11)( 8,10)(13,14)(15,18)(16,20)(17,19);
s2 := Sym(20)!( 3,16)( 4,15)( 5,17)( 6,19)( 7,18)( 8,20)( 9,13)(10,12)(11,14);
s3 := Sym(20)!( 6, 9)( 7,10)( 8,11);
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, s2*s3*s2*s3*s2*s3*s2*s3, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s3*s1*s2*s1*s2*s3*s2*s1*s2*s3*s1*s2*s1*s2*s3*s2*s1*s2 >; 
 

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