Janko group J4

In the area of modern algebra known as group theory, the Janko group J4 is a sporadic simple group of order
 * 86,775,571,046,077,562,880
 * = 221·33·5·7·113·23·29·31·37·43
 * ≈ 9.

History
J4 is one of the 26 Sporadic groups. Zvonimir Janko found J4 in 1975 by studying groups with an involution centralizer of the form 21 + 12.3.(M22:2). 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  210:SL5(2) and (210:24:A8):2 over a group 210:24:A8.

The Schur multiplier and the outer automorphism group are both trivial.

Since 37 and 43 are not supersingular primes, J4 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 and has rank 20, with point stabilizer of the form 211:M24. The 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
 * $$\begin{align}

a^2 &=b^3=c^2=(ab)^{23}=[a,b]^{12}=[a,bab]^5=[c,a]= \left ((ab)^2ab^{-1} \right)^3 \left (ab(ab^{-1})^2 \right)^3=\left (ab \left (abab^{-1} \right )^3 \right )^4 \\ &=\left [c,(ba)^2 b^{-1}ab^{-1} (ab)^3 \right]= \left (bc^{(bab^{-1}a)^2} \right )^3= \left ((bababab)^3 c c^{(ab)^3b(ab)^6b} \right )^2=1. \end{align}$$

Alternatively, one can start with the subgroup M24 and adjoin 3975 involutions, which are identified with the trios. By adding a certain relation, certain products of commuting involutions generate the binary Golay cocode, which extends to the maximal subgroup 211:M24. Bolt, Bray, and Curtis showed, using a computer, that adding just one more relation is sufficient to define J4.

Maximal subgroups
found the 13 conjugacy classes of maximal subgroups of J4 which are listed in the table below.

A Sylow 3-subgroup of J4 is a Heisenberg group: order 27, non-abelian, all non-trivial elements of order 3.