  The aim of this paper is to provide an atlas of identity bases for varieties
generated by small semigroups and groups. To help the working mathematician
easily find information, we provide a companion website that runs in the
background automated reasoning tools, finite model builders, and GAP, so that
the user has an automatic \textit{intelligent} guide on the literature.
  This paper is mainly a survey of what is known about identity bases for
semigroups or groups of small orders, and we also mend some gaps left
unresolved by previous authors. For instance, we provide the first complete and
justified list of identity bases for the varieties generated by a semigroup of
order up to~$4$, and the website contains the list of varieties generated by a
semigroup of order up to~$5$.
  The website also provides identity bases for several types of semigroups or
groups, such as bands, commutative groups, and metabelian groups. On the
inherently non-finitely based finite semigroups side, the website can decide if
a given finite semigroup possesses this property or not.
  We provide some other functionalities such as a tool that outputs the
multiplication table of a semigroup given by a $C$-presentation, where~$C$ is
any class of algebras defined by a set of first order formulas.
  The companion website can be found here
  \url{http://sgv.pythonanywhere.com}
  Please send any comments/suggestions to<EMAIL_ADDRESS>