  There are two purposes in this article. One is to present a criterion for the
existence of a birational embedding into a projective plane with inner and
outer Galois points for algebraic curves. Another is to classify plane curves
of degree $d$ admitting an inner Galois point $P$ and an outer Galois point $Q$
with $G_PG_Q=G_P \rtimes G_Q$ or $G_P \ltimes G_Q$, under the assumption that
the characteristic $p$ is zero or $p$ does not divide $d-1$.
