# Janko group J4

In the area of modern algebra known as group theory, the Janko group J4 is a sporadic simple group of order

221 · 33 ··· 113 · 23 · 29 · 31 · 37 · 43
= 86775571046077562880
≈ 9×1019.

## 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. 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.

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, 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

{\begin{aligned}a^{2}&=b^{3}=c^{2}=(ab)^{23}=[a,b]^{12}=[a,bab]^{5}=[c,a]=\left((ab)^{2}ab^{-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}cc^{(ab)^{3}b(ab)^{6}b}\right)^{2}=1.\end{aligned}} ## Maximal subgroups

Kleidman & Wilson (1988) found the 13 conjugacy classes of maximal subgroups of J4 as follows:

• 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.