# Janko group J3

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vte :
In the area of modern algebra known as group theory, the Janko group J$_3$ or the Higman-Janko-McKay group HJM is a sporadic simple group of order
    50,232,960 = 2$^7$ · 3$^5$ · 5 · 17 · 19.

## History and properties

J$_3$ is one of the 26 Sporadic groups and was predicted by Zvonimir Janko in 1969 as one of two new simple groups having 2$^{1+4}$:A$_5$ as a centralizer of an involution (the other is the Janko group J$_2$). J$_3$ was shown to exist by Graham Higman and John McKay (1969).
In 1982 R. L. Griess showed that J$_3$ cannot be a subquotient of the monster group. Thus it is one of the 6 sporadic groups called the pariahs.
J$_3$ has an outer automorphism group of order 2 and a Schur multiplier of order 3, and its triple cover has a unitary 9-dimensional representation over the finite field with 4 elements. Weiss (1982) constructed it via an underlying geometry. It has a modular representation of dimension eighteen over the finite field with 9 elements. It has a complex projective representation of dimension eighteen.

## Constructions
### Using matrices

J3 can be constructed by many different generators. Two from the ATLAS list are 18x18 matrices over the finite field of order 9, with matrix multiplication carried out with finite field arithmetic:
$\left({\begin{matrix}0&8&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\8&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\0&0&0&8&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\0&0&8&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\0&0&0&0&0&8&0&0&0&0&0&0&0&0&0&0&0&0\\0&0&0&0&8&0&0&0&0&0&0&0&0&0&0&0&0&0\\0&0&0&0&0&0&0&0&8&0&0&0&0&0&0&0&0&0\\0&0&0&0&0&0&0&0&0&8&0&0&0&0&0&0&0&0\\0&0&0&0&0&0&8&0&0&0&0&0&0&0&0&0&0&0\\0&0&0&0&0&0&0&8&0&0&0&0&0&0&0&0&0&0\\0&0&0&0&0&0&0&0&0&0&0&0&0&8&0&0&0&0\\0&0&0&0&0&0&0&0&0&0&0&0&0&0&8&0&0&0\\3&7&4&8&4&8&1&5&5&1&2&0&8&6&0&0&0&0\\0&0&0&0&0&0&0&0&0&0&8&0&0&0&0&0&0&0\\0&0&0&0&0&0&0&0&0&0&0&8&0&0&0&0&0&0\\0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&8\\4&8&6&2&4&8&0&4&0&8&4&5&0&8&1&1&8&5\\0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&8&0&0\\\end{matrix}}\right)$
and
$\left({\begin{matrix}4&8&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\0&0&8&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\4&4&8&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\0&0&0&0&8&0&0&0&0&0&0&0&0&0&0&0&0&0\\0&0&0&0&0&0&8&0&0&0&0&0&0&0&0&0&0&0\\0&0&0&0&0&0&0&8&0&0&0&0&0&0&0&0&0&0\\0&0&0&8&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\0&0&0&0&0&0&0&0&0&0&8&0&0&0&0&0&0&0\\0&0&0&0&0&0&0&0&0&0&0&8&0&0&0&0&0&0\\0&0&0&0&0&0&0&0&0&0&0&0&8&0&0&0&0&0\\0&0&0&0&0&8&0&0&0&0&0&0&0&0&0&0&0&0\\0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&8&0&0\\0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&8&0\\2&7&4&5&7&4&8&5&6&7&2&2&8&8&0&0&5&0\\4&7&5&8&6&1&1&6&5&3&8&7&5&0&8&8&6&0\\0&0&0&0&0&0&0&0&8&0&0&0&0&0&0&0&0&0\\0&0&0&0&0&0&0&0&0&8&0&0&0&0&0&0&0&0\\8&2&5&5&7&2&8&1&5&5&7&8&6&0&0&7&3&8\\\end{matrix}}\right)$

### Using the subgroup PSL(2,16)

The automorphism group J$_3$:2 can be constructed by starting with the subgroup PSL(2,16):4 and adjoining 120 involutions, which are identified with the Sylow 17-subgroups. Note that these 120 involutions are outer elements of J$_3$:2. One then defines the following relation:
$\left({\begin{matrix}1&1\\1&0\end{matrix}}\sigma t_{(\nu ,\nu 7)}\right)^{5}=1$
where $\sigma$ is the Frobenius automorphism or order 4, and $t_{(\nu ,\nu 7)}$ is the unique 17-cycle that sends
$\infty \rightarrow 0\rightarrow 1\rightarrow 7$
Curtis showed, using a computer, that this relation is sufficient to define J$_3$:2.

### Using a presentation

In terms of generators a, b, c, and d its automorphism group J$_3$:2 can be presented as $a^{17}=b^{8}=a^{b}a^{-2}=c^{2}=b^{c}b^{3}=(abc)^{4}=(ac)^{17}=d^{2}=[d,a]=[d,b]=(a^{3}b^{-3}cd)^{5}=1.$
A presentation for J$_3$ in terms of (different) generators a, b, c, d is $a^{19}=b^{9}=a^{b}a^{2}=c^{2}=d^{2}=(bc)^{2}=(bd)^{2}=(ac)^{3}=(ad)^{3}=(a^{2}ca^{-3}d)^{3}=1.$

## Maximal subgroups

Finkelstein & Rudvalis (1974) found the 9 conjugacy classes of maximal subgroups of J$_3$ as follows:
Maximal subgroups of J$_3$ :
| No. | Structure | Order | Index | Comments |
| 1 | L$_2$(16):2 | 8,160 = 2$^5$·3·5·17 | 6,156 = 2$^2$·3$^4$·19 | |
| 2,3 | L$_2$(19) | 3,420 = 2$^2$·3$^2$·5·19 | 14,688 = 2$^5$·3$^3$·17 | two classes, fused by an outer automorphism |
| 4 | 2$^4$: (3 × A$_5$) | 2,880 = 2$^6$·3$^2$·5 | 17,442 = 2·3$^3$·17·19 | |
| 5 | L$_2$(17) | 2,448 = 2$^4$·3$^2$·17 | 20,520 = 2$^3$·3$^3$·5·19 | centralizer of an outer automorphism of order 2 |
| 6 | (3 × A$_6$):2$_2$ | 2,160 = 2$^4$·3$^3$·5 | 23,256 = 2$^3$·3$^2$·17·19 | normalizer of a subgroup of order 3 (class 3A) |
| 7 | 3$^{2+1+2}$:8 | 1,944 = 2$^3$·3$^5$ | 25,840 = 2$^4$·5·17·19 | normalizer of a Sylow 3-subgroup |
| 8 | 2$^{1+4}$ $_–$:A$_5$ | 1,920 = 2$^7$·3·5 | 26,163 = 3$^4$·17·19 | centralizer of involution |
| 9 | 2$^{2+4}$: (3 × S$_3$) | 1,152 = 2$^7$·3$^2$ | 43,605 = 3$^3$·5·17·19 | |