Janko group J4
In the area of modern algebra known as group theory, the Janko group "J" is a sporadic simple group of order History. "J" is one of the 26 Sporadic groups. Zvonimir Janko found J in 1975 by studying groups with an involution centralizer of the form 2.3.(M:2). Its existence and uniqueness was shown using computer calculations by Simon P. Norton and others in 1980. It has a modular representation of dimension 112 over the finite field with 2 elements and is the stabilizer of a certain 4995 dimensional subspace of the exterior square, a fact which Norton  used to construct it, and which is the easiest way to deal with it computationally. and gave computer-free proofs of uniqueness. and gave a computer-free proof of existence by constructing it as an amalgams of groups 2:SL(2) and (2:2:A):2 over a group 2:2:A. The Schur multiplier and the outer automorphism group are both trivial. Since 37 and 43 are not supersingular primes, "J" cannot be a subquotient of the monster group. Thus it is one of the 6 sporadic groups called the pariahs. Representations. The smallest faithful complex representation has dimension 1333; there are  two complex conjugate representations of this dimension. The smallest faithful representation over any field is a 112 dimensional representation over the field of 2 elements. The smallest permutation representation is on 173067389 points, with point stabilizer of the form 2M. These points can be identified with certain "special vectors" in the 112 dimensional representation. Presentation. It has a presentation in terms of three generators a, b, and c as Maximal subgroups. found the 13 conjugacy classes of maximal subgroups of "J" as follows: A Sylow 3-subgroup is a Heisenberg group: order 27, non-abelian, all non-trivial elements of  order 3.