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# Chiral Polytope of Type {3,6,9}

Regular Cover :{3,6,9}*972a with group SmallGroup(972,108) = C33 ⋊ (D9 x C2)of order 972
Rank : 4
Schlafli Type : {3,6,9}
Rotation Group : SmallGroup(162,19) = (C33 ⋊ C3) ⋊ C2 of order 162
Number of vertices, edges, etc : 3, 9, 27, 9
If Aut({3,6,9}*972a)=<s0, s1, s2, s3>, then this chiral polytope is ({3,6,9}*972a)/N, where
N=<s0*s1*s2*s1*s0*s3*s2*s1*s3*s2*s1*s3> of order 3
Facet : Regular {3,6}*36
Vertex Figure : Chiral Quotient of {6,9}*324a
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, sig1^-1*sig2^-1*sig1^-1*sig2^-1, sig2^-1*sig3^-1*sig2^-1*sig3^-1,
sig1^-1*sig3*sig1*sig2*sig3*sig2^-1, sig1*sig2^-1*sig2^-1*sig2^-1*sig1*sig2^-1,
sig1*sig3^-1*sig1*sig3^-1*sig1*sig3^-1 ];;
rotpoly := F / rels;;

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

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

```