  We present an elementary derivation of the "intrinsic" symmetry groups for
knots and links of 8 or fewer crossings. The standard symmetry group for a link
is the mapping class group $\MCG(S^3,L)$ or $\Sym(L)$ of the pair $(S^3,L)$.
Elements in this symmetry group can (and often do) fix the link and act
nontrivially only on its complement. We ignore such elements and focus on the
"intrinsic" symmetry group of a link, defined to be the image $\Sigma(L)$ of
the natural homomorphism $\MCG(S^3,L) \rightarrow \MCG(S^3) \cross \MCG(L)$.
This different symmetry group, first defined by Whitten in 1969, records
directly whether $L$ is isotopic to a link $L'$ obtained from $L$ by permuting
components or reversing orientations.
  For hyperbolic links both $\Sym(L)$ and $\Sigma(L)$ can be obtained using the
output of \texttt{SnapPea}, but this proof does not give any hints about how to
actually construct isotopies realizing $\Sigma(L)$. We show that standard
invariants are enough to rule out all the isotopies outside $\Sigma(L)$ for all
links except $7^2_6$, $8^2_{13}$ and $8^3_5$ where an additional construction
is needed to use the Jones polynomial to rule out "component exchange"
symmetries. On the other hand, we present explicit isotopies starting with the
positions in Cerf's table of oriented links which generate $\Sigma(L)$ for each
link in our table. Our approach gives a constructive proof of the $\Sigma(L)$
groups.
