# Chiral Polytope of Type {8,4}

Regular Cover :{8,4}*1600b with group SmallGroup(1600,6690) = Q8 ⋊ (C52 ⋊ D4)of order 1600
Rank : 3
Schlafli Type : {8,4}
Rotation Group : SmallGroup(160,85) = C5 ⋊ ((C42) ⋊ C2) of order 160
Number of vertices, edges, etc : 40, 80, 20
If Aut({8,4}*1600b)=<s0, s1, s2>, then this chiral polytope is ({8,4}*1600b)/N, where
N=<s0*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s2> of order 5
Facet : (Regular) 8-gon
Vertex Figure : (Regular) 4-gon
Finitely Presented Group Representation of the Rotation Group(GAP) :
```F := FreeGroup("sig1","sig2");;
sig1 := F.1;;  sig2 := F.2;;
rels := [ sig1^-1*sig2^-1*sig1^-1*sig2^-1, sig2*sig2*sig2*sig2, sig1*sig1*sig1*sig1*sig1*sig1*sig1*sig1,
sig1*sig1*sig2^-1*sig1*sig1*sig1*sig2^-1*sig1, sig1^-1*sig2*sig1^-1*sig2*sig1*sig2^-1*sig1^-1*sig2*sig1*sig1*sig2^-1*sig2^-1 ];;
rotpoly := F / rels;;

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

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

```