Polytope of Type {3,2,6,12}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,2,6,12}*1296a
if this polytope has a name.
Group : SmallGroup(1296,2909)
Rank : 5
Schlafli Type : {3,2,6,12}
Number of vertices, edges, etc : 3, 3, 9, 54, 18
Order of s₀s₁s₂s₃s₄ : 12
Order of s₀s₁s₂s₃s₄s₃s₂s₁ : 2
Special Properties :
   Degenerate
   Universal
   Non-Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {3,2,6,4}*432
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := (2,3);;
s1 := (1,2);;
s2 := ( 7,10)( 8,11)( 9,12)(16,19)(17,20)(18,21);;
s3 := ( 5, 6)( 7, 8)(10,12)(13,19)(14,21)(15,20)(17,18);;
s4 := ( 4,13)( 5,15)( 6,14)( 7,16)( 8,18)( 9,17)(10,19)(11,21)(12,20);;
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, 
s1*s2*s1*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s0*s4*s0*s4, s1*s4*s1*s4, s2*s4*s2*s4, 
s0*s1*s0*s1*s0*s1, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s2*s3*s4*s3*s2*s3*s2*s3*s4*s2*s3*s2*s3, 
s2*s3*s4*s3*s4*s2*s3*s4*s3*s4*s2*s3*s4*s3*s4 ];;
poly := F / rels;;

Permutation Representation (Magma) :

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

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