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What is the difference between internal symmetries and space-time symmetries? Where would the $C$, $P$ and $T$ symmetries be classified?

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Space-time symmetries are symmetries with respect to coordinate transformations. In non-relativistic physics, space and time is assumed to be Galilean invariant. Galileo symmetries form a ten parameters continuous group. These parameters label three space translations, time translation, three rotations and three Galileo's boosts. It also includes discrete symmetries such as reflections. In relativistic physics, Galileo symmetry is replaced by Poincare symmetry, another ten parameter symmetry group generated by four translations, three rotations and three Lorentz boosts, and which also includes reflections.

On the other hand, internal symmetries refer to transformations acting on internal degrees of freedom of systems, such as fields, charges, etc, which does not transform space-time points. The classic example of internal symmetries are gauge symmetries. In quantum field theory, gauge symmetries are normally given by compact semi-simple Lie groups, such as the QCD's $SU(3)$ and the electroweak's $SU(2)\times U(1)$.

When people talk about internal symmetries, they normally refer to gauge symmetries. However (as remarked by AccidentalFourierTransform), gauge symmetries are not symmetries in a strict sense. A symmetry is a transformation relating distinct states of a given system which results in the same physics. Gauge symmetries is only a way of labeling the same state in different ways. It is therefore just a redundancy. See for instance: Gauge symmetry is not a symmetry?

Parity transformation (P) and time reversal (T) are coordinates reflections and therefore are space-time transformations. These transformations belong to Poincare group. Charge conjugation (C) acts only on charges, with no regards to space-time so it is an internal transformation.

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