I

found an old copy (Vol 2 Number 4 1980) of the The Mathematical Intelligencer with on its front

cover the list of the 26 _known_ sporadic groups together with a

starred added in proof saying

- added in

proof … the classification of finite simple groups is complete.

there are no other sporadic groups.

(click on the left picture to see a larger scanned image). In it is a

beautiful paper by John Conway “Monsters and moonshine” on the

classification project. Along the way he describes the simplest

non-trivial simple group $A_5 $ as the icosahedral group. as well as

other interpretations as Lie groups over finite fields. He also gives a

nice introduction to representation theory and the properties of the

character table allowing to reconstruct $A_5 $ only knowing that there

must be a simple group of order 60.

A more technical account

of the classification project (sketching the main steps in precise

formulations) can be found online in the paper by Ron Solomon On finite simple

groups and their classification. In addition to the posts by John Baez mentioned

in this

post he has a few more columns on Platonic solids and their relation to Lie

algebras, continued here.