Special relativity considers observers in inertial frames, and the covariance group consists of rotations, velocity boosts, and the parity transformation.
For example, the Maxwell equation with sources, transforms as a four-vector, that is, under the (1/2,1/2) representation of the O(1,3) group.
The Dirac equation, transforms as a bispinor, that is, under the (1/2,0)⊕(0,1/2) representation of the O(1,3) group.
For example, the Maxwell equation is invariant, while the corresponding equation for the weak field explicitly contains left currents and thus is not invariant under the parity transformation.
In general relativity the covariance group consists of all arbitrary (invertible and differentiable) coordinate transformations.