# Janko group J4

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In the area of modern algebra known as group theory, the Janko group J$_4$ is a sporadic simple group of order
    86,775,571,046,077,562,880
    = 2$^{21}$ · 3$^3$ · 5 · 7 · 11$^3$ · 23 · 29 · 31 · 37 · 43
    ≈ 9×10$^{19}$.

## History

J$_4$ is one of the 26 Sporadic groups. Zvonimir Janko found J$_4$ in 1975 by studying groups with an involution centralizer of the form 2$^{1+12}$.3.(M$_{22}$: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 2$^{10}$:SL$_5$(2) and (2$^{10}$:2$^4$:A$_8$):2 over a group 2$^{10}$:2$^4$:A$_8$.
The Schur multiplier and the outer automorphism group are both trivial.
Since 37 and 43 are not supersingular primes, J$_4$ 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 2$^{11}$:M$_{24}$. 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
    ${\begin{aligned}a^{2}&=b^{3}=c^{2}=(ab)^{23}=[a,b]^{12}=[a,bab]^{5}=[c,a]=\left((ab)^{2}ab^{-1}\right)^{3}\left(ab(ab^{-1})^{2}\right)^{3}=\left(ab\left(abab^{-1}\right)^{3}\right)^{4}\\&=\left[c,(ba)^{2}b^{-1}ab^{-1}(ab)^{3}\right]=\left(bc^{(bab^{-1}a)^{2}}\right)^{3}=\left((bababab)^{3}cc^{(ab)^{3}b(ab)^{6}b}\right)^{2}=1.\end{aligned}}$
Alternatively, one can start with the subgroup M$_{24}$ 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 2$^{11}$:M$_{24}$. Bolt, Bray, and Curtis showed, using a computer, that adding just one more relation is sufficient to define J$_4$.

## Maximal subgroups

Kleidman & Wilson (1988) found the 13 conjugacy classes of maximal subgroups of J$_4$ which are listed in the table below.
Maximal subgroups of J$_4$ :
| No. | Structure | Order | Index | Comments |
| 1 | 2$^{11}$:M$_{24}$ | 501,397,585,920 = 2$^{21}$·3$^3$·5·7·11·23 | 173,067,389 = 11$^2$·29·31·37·43 | contains a Sylow 2-subgroup and a Sylow 3-subgroup; contains the centralizer 2$^{11}$:(M$_{22}$:2) of involution of class 2B |
| 2 | 2$^{1+12}$ +$^·$3.(M$_{22}$:2) | 21,799,895,040 = 2$^{21}$·3$^3$·5·7·11 | 3,980,549,947 = 11$^2$·23·29·31·37·43 | centralizer of involution of class 2A; contains a Sylow 2-subgroup and a Sylow 3-subgroup |
| 3 | 2$^{10}$:L$_5$(2) | 10,239,344,640 = 2$^{20}$·3$^2$·5·7·31 | 8,474,719,242 = 2·3·11$^3$·23·29·37·43 | |
| 4 | 2$^{3+12·}$(S$_5$ × L$_3$(2)) | 660,602,880 = 2$^{21}$·3$^2$·5·7 | 131,358,148,251 = 3·11$^3$·23·29·31·37·43 | contains a Sylow 2-subgroup |
| 5 | U$_3$(11):2 | 141,831,360 = 2$^6$·3$^2$·5·11$^3$·37 | 611,822,174,208 = 2$^{15}$·3·7·23·29·31·43 | |
| 6 | M$_{22}$:2 | 887,040 = 2$^8$·3$^2$·5·7·11 | 97,825,995,497,472 = 2$^{13}$·3·11$^2$·23·29·31·37·43 | |
| 7 | 11$^{1+2}$ +:(5 × GL(2,3)) | 319,440 = 2$^4$·3·5·11$^3$ | 271,649,045,348,352 = 2$^{17}$·3$^2$·7·23·29·31·37·43 | normalizer of a Sylow 11-subgroup |
| 8 | L$_2$(32):5 | 163,680 = 2$^5$·3·5·11·31 | 530,153,782,050,816 = 2$^{16}$·3$^2$·7·11$^2$·23·29·37·43 | |
| 9 | PGL(2,23) | 12,144 = 2$^4$·3·11·23 | 7,145,550,975,467,520 = 2$^{17}$·3$^2$·5·7·11$^2$·29·31·37·43 | |
| 10 | U$_3$(3) | 6,048 = 2$^5$·3$^3$·7 | 14,347,812,672,962,560 = 2$^{16}$·5·11$^3$·23·29·31·37·43 | contains a Sylow 3-subgroup |
| 11 | 29:28 | 812 = 2$^2$·7·29 | 106,866,466,805,514,240 = 2$^{19}$·3$^3$·5·11$^3$·23·31·37·43 | Frobenius group; normalizer of a Sylow 29-subgroup |
| 12 | 43:14 | 602 = 2·7·43 | 144,145,466,853,949,440 = 2$^{20}$·3$^3$·5·11$^3$·23·29·31·37 | Frobenius group; normalizer of a Sylow 43-subgroup |
| 13 | 37:12 | 444 = 2$^2$·3·37 | 195,440,475,329,003,520 = 2$^{19}$·3$^2$·5·7·11$^3$·23·29·31·43 | Frobenius group; normalizer of a Sylow 37-subgroup |
A Sylow 3-subgroup of J$_4$ is a Heisenberg group: order 27, non-abelian, all non-trivial elements of order 3.