In the area of modern algebra known as group theory, the Janko group J<sub>4</sub> is a sporadic simple group of order
J<sub>4</sub> is one of the 26 Sporadic groups. Zvonimir Janko found J<sub>4</sub> in 1975 by studying groups with an involution centralizer of the form 2<sup>1 + 12</sup>.3.(M<sub>22</sub>: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<sup>11</sup>:M<sub>24</sub>, 2<sup>10</sup>:L<sub>5</sub>(2), and 2<sup>3+12</sup>.(L<sub>3</sub>(2)xS<sub>5</sub>) over their intersections.
The Schur multiplier and the outer automorphism group are both trivial.
Since 37 and 43 are not supersingular primes, J<sub>4</sub> cannot be a subquotient of the monster group. Thus it is one of the 6 sporadic groups called the pariahs.
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=11<sup>2</sup>29313743 points, with point stabilizer of the form 2<sup>11</sup>:M<sub>24</sub>. This permutation representation has rank 7; the suborbit lengths are 1, 15180=2<sup>2</sup>351123, 28336=2<sup>4</sup>71123, 3400320=2<sup>7</sup>3571123, 32643072=2<sup>11</sup>3<sup>2</sup>71123, 54405120=2<sup>11</sup>3571123, and 82575360=2<sup>18</sup>3<sup>2</sup>7. The points can be identified with certain "special vectors" in the 112 dimensional representation.
The degrees of irreducible representations of the Janko group J<sub>4</sub> are 1, 1333, 1333, 299367, 299367, ... .
It has a presentation in terms of three generators a, b, and c as
Alternatively, one can start with the subgroup M<sub>24</sub> and adjoin 3975 involutions, which are identified with the trios. By adding a certain relation, certain products of commuting involutions generate the binary Golay code, which extends to the maximal subgroup 2<sup>11</sup>:M<sub>24</sub>. Bolt, Bray, and Curtis showed, using a computer, that adding just one more relation is sufficient to define J<sub>4</sub>.
found the 13 conjugacy classes of maximal subgroups of J<sub>4</sub> which are listed in the table below.
A Sylow 3-subgroup of J<sub>4</sub> is a Heisenberg group: order 27, non-abelian, all non-trivial elements of order 3.