The Janko groups denoted by $J_1, J_2, J_3$ and $J_4$ are four of the 26 sporadic groups. They were discovered by Z. Janko in 1966 and published in the article "A new finite simple group with abelian Sylow $2$ subgroups and its characterization.'' (Journal of Algebra, 3, 1966, 32: 147-186).

Each of these groups have very intricate matrix representations as maps into large general linear groups. For example, the matrix $K$ corresponding to $J_4$ gives a representation of $J_4$ in $GL_{112}(2)$