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# Polytope of Type {5,4,2}

Atlas Canonical Name : {5,4,2}*1440
if this polytope has a name.
Group : SmallGroup(1440,5843)
Rank : 4
Schlafli Type : {5,4,2}
Number of vertices, edges, etc : 90, 180, 72, 2
Order of s0s1s2s3 : 8
Order of s0s1s2s3s2s1 : 2
Special Properties :
Degenerate
Universal
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) :
No Regular Quotients.
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
```s0 := ( 1, 3)( 2, 8)( 4,10)( 5, 6)( 7, 9);;
s1 := ( 1, 2)( 3, 4)( 5, 9)( 6, 8)( 7,10);;
s2 := ( 1, 2)( 3, 8)( 4, 7)( 5, 6)( 9,10);;
s3 := (11,12);;
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,
s1*s2*s1*s2*s1*s2*s1*s2, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1 ];;
poly := F / rels;;

```
Permutation Representation (Magma) :
```s0 := Sym(12)!( 1, 3)( 2, 8)( 4,10)( 5, 6)( 7, 9);
s1 := Sym(12)!( 1, 2)( 3, 4)( 5, 9)( 6, 8)( 7,10);
s2 := Sym(12)!( 1, 2)( 3, 8)( 4, 7)( 5, 6)( 9,10);
s3 := Sym(12)!(11,12);
poly := sub<Sym(12)|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, s1*s2*s1*s2*s1*s2*s1*s2,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1 >;

```

to this polytope