  We study the noncommutative Riemannian geometry of the alternating group
$A_4=(Z_2 \times Z_2)\cross Z_3$ using a recent formulation for finite groups.
We find a unique `Levi-Civita' connection for the invariant metric, and find
that it has Ricci-flat but nonzero Riemann curvature. We show that it is the
unique Ricci-flat connection on $A_4$ with the standard framing (we solve the
vacuum Einstein's equation). We also propose a natural Dirac operator for the
associated spin connection and solve the Dirac equation. Some of our results
hold for any finite group equipped with a cyclic conjugacy class of 4 elements.
In this case the exterior algebra $\Omega(A_4)$ has dimensions
$1:4:8:11:12:12:11:8:4:1$ with top-form 9-dimensional. We also find the
noncommutative cohomology $H^1(A_4)=C$.
