I have often seen shorthand notation used in group-theoretic contexts and I believe it is called ATLAS notation. However, even with some searching I have not been able to find a satisfactory summary of the meaning of this notation.
I have found the meaning of some things, like $A \times B$ for the direct product of $A$ and $B$, $G=A.B$ for a group with a normal subgroup isomorphic to $A$ and for which $G/A \cong B$, $A:B$ when $A.B$ is a split extension, and $A \cdot B$ when $A.B$ is not split.
However, I have seen things like $4 \times 4 \times 4$ and $3^3.\mathrm{SL}_3(3)$. What do numbers by themselves, possibly with powers, represent - the cyclic groups of those orders, perhaps?
What is a good, complete reference for this notation, available freely online? If no such thing exists, a nice explanation in an answer would also work.
Thanks.
EDIT: Here is an example of a paper where I have seen this notation.