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# Polytope of Type {4,6,4}

Atlas Canonical Name : {4,6,4}*720a
if this polytope has a name.
Group : SmallGroup(720,763)
Rank : 4
Schlafli Type : {4,6,4}
Number of vertices, edges, etc : 10, 45, 45, 6
Order of s0s1s2s3 : 6
Order of s0s1s2s3s2s1 : 4
Special Properties :
Non-Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{4,6,4,2} of size 1440
Vertex Figure Of :
{2,4,6,4} of size 1440
Quotients (Maximal Quotients in Boldface) :
No Regular Quotients.
Covers (Minimal Covers in Boldface) :
2-fold covers : {4,6,4}*1440a, {4,6,4}*1440b, {4,6,4}*1440c
Permutation Representation (GAP) :
s0 := (4,6);;
s1 := (3,4)(5,6);;
s2 := (2,3);;
s3 := (1,2)(3,5)(4,6);;
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, s0*s1*s0*s1*s0*s1*s0*s1,
s2*s3*s2*s3*s2*s3*s2*s3, s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2,
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s1*s2*s3*s2*s1*s2*s1*s2*s3*s1*s2*s1*s2,
s1*s2*s3*s2*s1*s3*s0*s1*s2*s3*s2*s0*s1*s0 ];;
poly := F / rels;;

Permutation Representation (Magma) :
s0 := Sym(6)!(4,6);
s1 := Sym(6)!(3,4)(5,6);
s2 := Sym(6)!(2,3);
s3 := Sym(6)!(1,2)(3,5)(4,6);
poly := sub<Sym(6)|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,
s0*s1*s0*s1*s0*s1*s0*s1, s2*s3*s2*s3*s2*s3*s2*s3,
s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2,
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s1*s2*s3*s2*s1*s2*s1*s2*s3*s1*s2*s1*s2,
s1*s2*s3*s2*s1*s3*s0*s1*s2*s3*s2*s0*s1*s0 >;

References : None.
to this polytope