  We describe the collection of finite simple groups, with a view on physical
applications. We recall first the prime cyclic groups $Z_p$, and the
alternating groups $Alt_{n>4}$. After a quick revision of finite fields
$\mathbb{F}_q$, $q = p^f$, with $p$ prime, we consider the 16 families of
finite simple groups of Lie type. There are also 26 \emph{extra} "sporadic"
groups, which gather in three interconnected "generations" (with 5+7+8 groups)
plus the Pariah groups (6). We point out a couple of physical applications,
including constructing the biggest sporadic group, the "Monster" group, with
close to $10^{54}$ elements from arguments of physics, and also the relation of
some Mathieu groups with compactification in string and M-theory.
