# Janko group J2

Algebraic structure → Group theory Group theory :
Basic notions
Subgroup
Normal subgroup
Group action
Quotient group
(Semi-)direct product
Direct sum
Free product
Wreath product
Group homomorphisms
kernel
image
simple
finite
infinite
continuous
multiplicative
additive
cyclic
abelian
dihedral
nilpotent
solvable
Glossary of group theory
List of group theory topics :
Finite groups
Cyclic group Z$_n$
Symmetric group S$_n$
Alternating group A$_n$
Dihedral group D$_n$
Quaternion group Q
Cauchy's theorem
Lagrange's theorem
Sylow theorems
Hall's theorem
p-group
Elementary abelian group
Frobenius group
Schur multiplier
Classification of finite simple groups
cyclic
alternating
Lie type
sporadic :
Discrete groupsLattices
Integers ($\mathbb {Z}$)
Free group
Modular groups PSL(2, $\mathbb {Z}$)SL(2, $\mathbb {Z}$)
Arithmetic group
Lattice
Hyperbolic group :
Topological and Lie groups
Solenoid
Circle
General linear GL(n)
Special linear SL(n)
Orthogonal O(n)
Euclidean E(n)
Special orthogonal SO(n)
Unitary U(n)
Special unitary SU(n)
Symplectic Sp(n)
G$_2$
F$_4$
E$_6$
E$_7$
E$_8$
Lorentz
Poincaré
Conformal
Diffeomorphism
Loop
Infinite dimensional Lie group O(∞)SU(∞)Sp(∞) :
Algebraic groups
Linear algebraic group
Reductive group
Abelian variety
Elliptic curve :
vte :
In the area of modern algebra known as group theory, the Janko group J$_2$ or the Hall-Janko group HJ is a sporadic simple group of order
    604,800 = 2$^7$ · 3$^3$ · 5$^2$ · 7
    ≈ 6×10$^5$.

## History and properties

J$_2$ is one of the 26 Sporadic groups and is also called Hall–Janko–Wales group. In 1969 Zvonimir Janko predicted J$_2$ 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 J3). It was constructed by Marshall Hall and David Wales (1968) as a rank 3 permutation group on 100 points.
Both the Schur multiplier and the outer automorphism group have order 2. As a permutation group on 100 points J$_2$ has involutions moving all 100 points and involutions moving just 80 points. The former involutions are products of 25 double transportions, an odd number, and hence lift to 4-elements in the double cover 2.A$_{100}$. The double cover 2.J$_2$ occurs as a subgroup of the Conway group Co$_0$.
J$_2$ is the only one of the 4 Janko groups that is a subquotient 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 is also a subgroup of index two of the group of automorphisms of the Hall–Janko Near Octagon, leading to a permutation representation of degree 315.
It has a modular representation of dimension six over the field of four elements; if in characteristic two we have w$^2$ + w + 1 = 0, then J$_2$ is generated by the two matrices
    ${\mathbf {A} }={\begin{pmatrix}w^{2}&w^{2}&0&0&0&0\\1&w^{2}&0&0&0&0\\1&1&w^{2}&w^{2}&0&0\\w&1&1&w^{2}&0&0\\0&w^{2}&w^{2}&w^{2}&0&w\\w^{2}&1&w^{2}&0&w^{2}&0\end{pmatrix}}$
and
    ${\mathbf {B} }={\begin{pmatrix}w&1&w^{2}&1&w^{2}&w^{2}\\w&1&w&1&1&w\\w&w&w^{2}&w^{2}&1&0\\0&0&0&0&1&1\\w^{2}&1&w^{2}&w^{2}&w&w^{2}\\w^{2}&1&w^{2}&w&w^{2}&w\end{pmatrix}}.$
These matrices satisfy the equations
    ${\mathbf {A} }^{2}={\mathbf {B} }^{3}=({\mathbf {A} }{\mathbf {B} })^{7}=({\mathbf {A} }{\mathbf {B} }{\mathbf {A} }{\mathbf {B} }{\mathbf {B} })^{12}=1.$
(Note that matrix multiplication on a finite field of order 4 is defined slightly differently from ordinary matrix multiplication. See Finite field § Field with four elements for the specific addition and multiplication tables, with w the same as a and w$^2$ the same as 1 + a.)
J$_2$ 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 G$_2$(4). There is only one conjugacy class of J$_2$ in G$_2$(4). Every subgroup J$_2$ contained in G$_2$(4) extends to a subgroup J$_2$:2 = Aut(J$_2$) in G$_2$(4):2 = Aut(G$_2$(4)) (G$_2$(4) extended by the field automorphisms of F$_4$). G$_2$(4) is in turn isomorphic to a subgroup of the Conway group Co$_1$.

## Maximal subgroups

There are 9 conjugacy classes of maximal subgroups of J$_2$. Some are here described in terms of action on the Hall–Janko graph.
Maximal subgroups of J$_2$ :
| No. | Structure | Order | Index | Comments |
| 1 | U$_3$(3) | 6,048 = 2$^5$·3$^3$·7 | 100 = 2$^2$·5$^2$ | one-point stabilizer, with orbits of sizes 1, 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.S$_4$, which contains 6 additional involutions. |
| 2 | 3.PGL$_2$(9) | 2,160 = 2$^4$·3$^3$·5 | 280 = 2$^3$·5·7 | normalizer of a subgroup of order 3 (class 3A); has a subquotient A$_6$ |
| 3 | 2$^{1+4}$ $_–$:A$_5$ | 1,920 = 2$^7$·3·5 | 315 = 3$^2$·5·7 | centralizer of involution of class 2A moving 80 points |
| 4 | 2$^{2+4}$:(3 × S$_3$) | 1,152 = 2$^7$·3$^2$ | 525 = 3·5$^2$·7 | |
| 5 | A$_4$ × A$_5$ | 720 = 2$^4$·3$^2$·5 | 840 = 2$^3$·3·5·7 | contains 2$^2$ × A$_5$ (of order 240), centralizer of 3 involutions each moving 100 points |
| 6 | A$_5$ × D$_{10}$ | 600 = 2$^3$·3·5$^2$ | 1,008 = 2$^4$·3$^2$·7 | normalizer of a subgroup of order 5 (classes 5A, 5B) |
| 7 | PGL$_2$(7) | 336 = 2$^4$·3·7 | 1,800 = 2$^3$·3$^2$·5$^2$ | |
| 8 | 5$^2$:D$_{12}$ | 300 = 2$^2$·3·5$^2$ | 2,016 = 2$^5$·3$^2$·7 | |
| 9 | A$_5$ | 60 = 2$^2$·3·5 | 10,080 = 2$^5$·3$^2$·5·7 | |

## 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 = 3$^2$ · 5 · 7 | 2$^{40}$, 1 class |
| 2 = 2 | 2520 = 2$^3$ · 3$^2$ · 5 · 7 | 2$^{50}$, 1 class |
| 3 = 3 | 560 = 2$^4$ · 5 · 7 | 3$^{30}$, 1 class |
| 3 = 3 | 16800 = 2$^5$ · 3 · 5$^2$ · 7 | 3$^{32}$, 1 class |
| 4 = 2$^2$ | 6300 = 2$^2$ · 3$^2$ · 5$^2$ · 7 | 2$^6$4$^{20}$, 1 class |
| 5 = 5 | 4032 = 2$^6$ · 3$^2$ · 7 | 5$^{20}$, 2 classes, power equivalent |
| 5 = 5 | 24192 = 2$^7$ · 3$^3$ · 7 | 5$^{20}$, 2 classes, power equivalent |
| 6 = 2 · 3 | 25200 = 2$^4$ · 3$^2$ · 5$^2$ · 7 | 2$^4$3$^6$6$^{12}$, 1 class |
| 6 = 2 · 3 | 50400 = 2$^5$ · 3$^2$ · 5$^2$ · 7 | 2$^2$6$^{16}$, 1 class |
| 7 = 7 | 86400 = 2$^7$ · 3$^3$ · 5$^2$ | 7$^{14}$, 1 class |
| 8 = 2$^3$ | 75600 = 2$^4$ · 3$^3$ · 5$^2$ · 7 | 2$^3$4$^3$8$^{10}$, 1 class |
| 10 = 2 · 5 | 60480 = 2$^6$ · 3$^3$ · 5 · 7 | 10$^{10}$, 2 classes, power equivalent |
| 10 = 2 · 5 | 120960 = 2$^7$ · 3$^3$ · 5 · 7 | 5$^4$10$^8$, 2 classes, power equivalent |
| 12 = 2$^2$ · 3 | 50400 = 2$^5$ · 3$^2$ · 5$^2$ · 7 | 3$^2$4$^2$6$^2$12$^6$, 1 class |
| 15 = 3 · 5 | 80640 = 2$^8$ · 3$^2$ · 5 · 7 | 5$^2$15$^6$, 2 classes, power equivalent |