Groups are a mathematical tool used for measuring symmetry. As such, they play a fundamental role not just in mathematics, but also in physics, chemistry, encryption and coding theory. The theory of finite groups culminated in 2004, with the achievement of the classification of finite simple groups (CFSG), which is of the irreducible elements of the theory. Since the end of the 1950s, the CFSG has involved hundreds of mathematicians (including the Fields medal awardees J. Thompson and E. Bombieri, and the Cole Prize awardees W. Feit and M. Aschbacher). The scientific importance of the CFSG, its size (over 15,000 pages), its complexity and its demonstration have immediately made evident the need of a revision and reorganisation work. Currently, there are three ongoing international projects for this revision, with two of which we are actively cooperating. Our research concerns, in particular, a few large sporadic groups, which are the 26 exceptional cases of the CFSG. Together with C. Franchi of the University Cattolica and R. Solomon of the Ohio State University, we gave a local characterisation of the groups of Lyons and Harada-Norton, in line with a project started by D. Gorenstein and R. Lyons for an intrinsic characterisation of the large sporadic groups, instrumental for the revision project coordinated by Lyons and R. Solomon. Together with S. Shpectorov and C. Parker, of the University of Birmingham, and Franchi, we are studying a local geometry associate with the Thompson group, in order to provide a computer-free demonstration of its uniqueness. Finally, together with A.A. Ivanov of the Imperial College and Franchi we have an ongoing project for studying the representations of the Monster Group, the largest of the sporadic groups (and, in the words of Sir M. Atyiah, another Fields Medal awardee, “the most interesting outcome of the classification”). The Monster can be realised as a group of symmetries of the Griess algebra, which is an Euclidean commutative non-associative algebra of dimension 196884, corresponding to the degree 2 piece of the Monster algebra, a vertex operator algebra (VOA) construed in 1988 by I. Fraenkel, J. Lepowsky and A. Meurman in order to study the Monstrous Moonshine conjecture (a conjecture later solved, in 1992, by R. E. Borcherds who was awarded the Fields Medal for this feat). In 1996 M. Miyamoto demonstrated that, in general, the degree 2 components of the real VOAs admit idempotent generators (the Ising vectors) to which particular involutional automorphisms are associate (Miyamoto involutions). These, in the case of the Griess algebra, coincide with the involution class 2A of the Monster (that is, with the involutions having as their centraliser the double covering of the Baby Monster). In 2007 S. Sakuma demonstrated that, in general, an algebra generated by two Ising vectors is isomorphic with one of the nine dihedral subalgebras of the Griess algebra, classified by S. Norton in 1996. This result brought Ivanov to introduce in 2009 the abstract concepts of the Majorana algebra and of the Majorana representation, in order to provide an axiomatic structure that is independent from the VOAs (which, for the time being, are only very partially understood), for studying the Monster. The name comes from the fact that the fusion rules of these algebras are the same as those of the VOAs that are associate with the two-dimensional Ising model, that is, to the lattice model of free Majorana fermions. Within this framework, we have achieved important results concerning the Majorana representations of the group of Harada-Norton and of symmetric groups. One of the major difficulties in this field of study is due to the dimensions of the degrees of these representations, making the traditional methods of linear algebra unusable. In order to overcome this obstacle, we used algebraic combinatorics methods (association schemes), expanding some important concepts and outcomes thereof (first eigenmatrix, orthogonality relations). Moreover, the research group has been organising, for more than ten years, a summer school on finite simple groups, which, every other year, attracts participants from universities from all over the world (including Oxford, Cambridge, the Imperial College, Birmingham, Warwick, Halle-Wittenberg, Western Australia, Caltech and Southern California). The school has been a springboard for the career of many of them.