"Many of the finite simple groups can be described as symmetries of finite geometries, and it remains a hot topic in group theory to expand our knowledge of the Classification of Finite Simple Groups using finite geometry."
— Finite geometry page at the Centre for the Mathematics of
Symmetry and Computation at the University of Western Australia
(Alice Devillers, John Bamberg, Gordon Royle)
For such symmetries, see Robert A. WIlson's recent book The Finite Simple Groups.
The finite simple groups are often described as the "building blocks" of finite group theory.
At least some of these building blocks have their own building blocks. See Non-Euclidean Blocks.
For instance, a set of 24 such blocks (or, more simply, 24 unit squares) appears in the Miracle Octad Generator (MOG) of R.T. Curtis, used in the study of the finite simple group M24.
(The octads of the MOG illustrate yet another sort of mathematical blocks— those of a block design.)