# Hall–Janko group

In mathematics, the Hall–Janko group HJ, is a finite simple sporadic group of order 604800. It is also called the second Janko group J2, or the Hall-Janko-Wales group, since it was predicted by Janko (1969) as one of two new groups with an involution centralizer of the form 21+4A5 (the other is the Janko group J3) and constructed by Hall and Wales (1968) as a rank 3 permutation group on 100 points.

J2 is the only one of the 4 Janko groups that is a section of the Monster group; it is thus part of what Robert Griess calls the Happy Family. Since it is also found in the Conway group Co1, it is therefore part of the second generation of the Happy Family.

## Representations

It is a subgroup of index two of the group of automorphisms of the Hall-Janko graph, leading to a permutation representation of degree 100.

It has a modular representation of dimension six over the field of four elements; if in characteristic two we have w2 + w + 1 = 0, then J2 is generated by the two matrices

${\mathbf A} = \left ( \begin{matrix} w^2 & w^2 & 0 & 0 & 0 & 0 \\ 1 & w^2 & 0 & 0 & 0 & 0 \\ 1 & 1 & w^2 & w^2 & 0 & 0 \\ w & 1 & 1 & w^2 & 0 & 0 \\ 0 & w^2 & w^2 & w^2 & 0 & w \\ w^2 & 1 & w^2 & 0 & w^2 & 0 \end{matrix} \right )$

and

${\mathbf B} = \left ( \begin{matrix} w & 1 & w^2 & 1 & w^2 & w^2 \\ w & 1 & w & 1 & 1 & w \\ w & w & w^2 & w^2 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 & 1 \\ w^2 & 1 & w^2 & w^2 & w & w^2 \\ w^2 & 1 & w^2 & w & w^2 & w \end{matrix} \right ).$

These matrices satisfy the equations

${\mathbf A}^2 = {\mathbf B}^3 = ({\mathbf A}{\mathbf B})^7 = ({\mathbf A}{\mathbf B}{\mathbf A}{\mathbf B}{\mathbf B})^{12} = 1.$

J2 is thus a Hurwitz group, a finite homomorphic image of the (2,3,7) triangle group.

The matrix representation given above constitutes an embedding into Dickson's group G2(4). There are two conjugacy classes of HJ in G2(4), and they are equivalent under the automorphism on the field F4. Their intersection (the "real" subgroup) is simple of order 6048. G2(4) is in turn isomorphic to a subgroup of the Conway group Co1.

## Maximal subgroups

J2 has 9 conjugacy classes of maximal subgroups. Some are here described in terms of action on the Hall-Janko graph.

• U3(3) order 6048 - one-point stabilizer, with orbits of 36 and 63
Simple, containing 36 simple subgroups of order 168 and 63 involutions, all conjugate, each moving 80 points. A given involution is found in 12 168-subgroups, thus fixes them under conjugacy. Its centralizer has structure 4.S4, which contains 6 additional involutions.
• 3.PGL(2,9) order 2160 - has a subquotient A6
• 21+4:A5 order 1920 - centralizer of involution moving 80 points
• 22+4:(3 × S3) order 1152
• A4 × A5 order 720
Containing 22 × A5 (order 240), centralizer of 3 involutions each moving 100 points
• A5 × D10 order 600
• PGL(2,7) order 336
• 52:D12 order 300
• A5 order 60

## Conjugacy classes

The maximum order of any element is 15. As permutations, elements act on the 100 vertices of the Hall-Janko graph.

Order No. elements Cycle structure and conjugacy
1 = 1 1 = 1 1 class
2 = 2 315 = 32 · 5 · 7 240, 1 class
2520 = 23 · 32 · 5 · 7 250, 1 class
3 = 3 560 = 24 · 5 · 7 330, 1 class
16800 = 25 · 3 · 52 · 7 332, 1 class
4 = 22 6300 = 22 · 32 · 52 · 7 26420, 1 class
5 = 5 4032 = 26 · 32 · 7 520, 2 classes, power equivalent
24192 = 27 · 33 · 7 520, 2 classes, power equivalent
6 = 2 · 3 25200 = 24 · 32 · 52 · 7 2436612, 1 class
50400 = 25 · 32 · 52 · 7 22616, 1 class
7 = 7 86400 = 27 · 33 · 52 714, 1 class
8 = 23 75600 = 24 · 33 · 52 · 7 2343810, 1 class
10 = 2 · 5 60480 = 26 · 33 · 5 · 7 1010, 2 classes, power equivalent
120960 = 27 · 33 · 5 · 7 54108, 2 classes, power equivalent
12 = 22 · 3 50400 = 25 · 32 · 52 · 7 324262126, 1 class
15 = 3 · 5 80640 = 28 · 32 · 5 · 7 52156, 2 classes, power equivalent

## References

• Robert L. Griess, Jr., "Twelve Sporadic Groups", Springer-Verlag, 1998.
• Hall, Marshall; Wales, David (1968), "The simple group of order 604,800", Journal of Algebra 9: 417–450, doi:10.1016/0021-8693(68)90014-8, ISSN 0021-8693, MR0240192 (Griess relates [p. 123] how Marshall Hall, as editor of The Journal of Algebra, received a very short paper entitled "A simple group of order 604801." Yes, 604801 is prime.)
• Janko, Zvonimir (1969), "Some new simple groups of finite order. I", Symposia Mathematica (INDAM, Rome, 1967/68), Vol. 1, Boston, MA: Academic Press, pp. 25–64, MR0244371
• Wales, David B., "The uniqueness of the simple group of order 604800 as a subgroup of SL(6,4)", Journal of Algebra 11 (1969), 455 - 460.
• Wales, David B., "Generators of the Hall-Janko group as a subgroup of G2(4)", Journal of Algebra 13 (1969), 513–516, doi:10.1016/0021-8693(69)90113-6, MR0251133, ISSN 0021-8693