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# Chiral Polytope of Type {27,6}

Regular Cover :{27,6}*972a with group SmallGroup(972,107) = (C27 x C3) ⋊ D6of order 972
Rank : 3
Schlafli Type : {27,6}
Rotation Group : SmallGroup(162,9) = (C27 ⋊ C3) ⋊ C2 of order 162
Number of vertices, edges, etc : 27, 81, 6
If Aut({27,6}*972a)=<s0, s1, s2>, then this chiral polytope is ({27,6}*972a)/N, where
N=<s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s2*s1*s0*s1*s0*s2*s1*s0*s2*s1*s2> of order 3
Facet : (Regular) 27-gon
Vertex Figure : (Regular) 6-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, sig1*sig1*sig2^-1*sig1*sig1*sig2^-1, sig2*sig2*sig2*sig2*sig2*sig2,
sig1^-1*sig2*sig1^-1*sig2^-1*sig2^-1*sig1*sig2^-1*sig2^-1*sig1^-1*sig2, sig1*sig1*sig1*sig1*sig1*sig2*sig1^-1*sig1^-1*sig1^-1*sig2*sig2*sig1^-1*sig2 ];;
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, sig1*sig1*sig2^-1*sig1*sig1*sig2^-1,
sig2*sig2*sig2*sig2*sig2*sig2, sig1^-1*sig2*sig1^-1*sig2^-1*sig2^-1*sig1*sig2^-1*sig2^-1*sig1^-1*sig2,
sig1*sig1*sig1*sig1*sig1*sig2*sig1^-1*sig1^-1*sig1^-1*sig2*sig2*sig1^-1*sig2 >;

```