# Janko group J4

In mathematics, the fourth Janko group J4 is the sporadic finite simple group of order

221 · 33 ··· 113 · 23 · 29 · 31 · 37 · 43 (= 86775571046077562880)

whose existence was suggested by Janko (1976).

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

J4 is one of the 6 sporadic simple groups known as the "pariah groups" as they are not found within the Monster group. The order of the monster group is not divisible by 37 or 43.

## Existence and uniqueness

Janko found J₄ 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 of two 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. Aschbacher & Segev (1991) and Ivanov (1992) gave computer-free proofs of uniqueness. Ivanov & Meierfrankenfeld (1999) and Ivanov (2004) 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.

## 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 211M24. 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

$a^2=b^3=c^2=(ab)^{23}=[a,b]^{12}=[a,bab]^5=[c,a]=$
$(ababab^{-1})^3(abab^{-1}ab^{-1})^3=(ab(abab^{-1})^3)^4=$
$[c,bab(ab^{-1})^2(ab)^3]=(bc^{bab^{-1}abab^{-1}a})^3=$
$((bababab)^3cc^{(ab)^3b(ab)^6b})^2=1.$

## Maximal subgroups

Kleidman & Wilson (1988) showed that J4 has 13 conjugacy classes of maximal subgroups.

• 211:M24 - containing Sylow 2-subgroups and Sylow 3-subgroups; also containing 211:(M22:2), centralizer of involution of class 2B
• 21+12.3.(M22:2) - centralizer of involution of class 2A - containing Sylow 2-subgroups and Sylow 3-subgroups
• 210:PSL(5,2)
• 23+12.(S5 × PSL(3,2)) - containing Sylow 2-subgroups
• U3(11):2
• M22:2
• 111+2:(5 × GL(2,3)) - normalizer of Sylow 11-subgroup
• PSL(2,32):5
• PGL(2,23)
• U3(3) - containing Sylow 3-subgroups
• 29:28 Frobenius group
• 43:14 Frobenius group
• 37:12 Frobenius group

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