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# Chiral Polytope of Type {3,3,8}

Regular Cover :{3,3,8}*768b with group SmallGroup(768,1086052) = ((Q8 x C2) ⋊ C2) ⋊ S4of order 768
Rank : 4
Schlafli Type : {3,3,8}
Rotation Group : SmallGroup(192,1494) = Q8 ⋊ S4 of order 192
Number of vertices, edges, etc : 4, 12, 32, 16
If Aut({3,3,8}*768b)=<s0, s1, s2, s3>, then this chiral polytope is ({3,3,8}*768b)/N, where
N=<s0*s1*s0*s2*s1*s3*s2*s1*s0*s3*s2*s1*s3*s2*s3*s2> of order 2
Facet : Regular {3,3}*24
Vertex Figure : Regular {3,8}*96
Finitely Presented Group Representation of the Rotation Group(GAP) :
```F := FreeGroup("sig1","sig2","sig3");;
sig1 := F.1;;  sig2 := F.2;;  sig3 := F.3;;
rels := [ sig1*sig1*sig1, sig2*sig2*sig2, sig1^-1*sig2^-1*sig1^-1*sig2^-1, sig2^-1*sig3^-1*sig2^-1*sig3^-1,
sig1*sig2^-1*sig3*sig1^-1*sig3^-1, sig1*sig3*sig3*sig2*sig3^-1*sig1*sig3*sig2^-1 ];;
rotpoly := F / rels;;

```
Finitely Presented Group Representation of the Rotation Group (Magma) :

```rotpoly<sig1,sig2,sig3> := Group< sig1,sig2,sig3 | sig1*sig1*sig1, sig2*sig2*sig2,
sig1^-1*sig2^-1*sig1^-1*sig2^-1, sig2^-1*sig3^-1*sig2^-1*sig3^-1, sig1*sig2^-1*sig3*sig1^-1*sig3^-1,
sig1*sig3*sig3*sig2*sig3^-1*sig1*sig3*sig2^-1 >;

```