Janko group J2


In the area of modern algebra known as group theory, the Janko group J2 or the Hall-Janko group HJ is a sporadic simple group of order

History and properties

J2 is one of the 26 Sporadic groups and is also called Hall–Janko–Wales group. In 1969 Zvonimir Janko predicted J2 as one of two new simple groups having 21+4:A5 as a centralizer of an involution. It was constructed by 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 J2 has involutions moving all 100 points and involutions moving just 80 points. The former involutions are
products of 25 double transportions, an odd number number, and hence lift to 4-elements in the double cover 2.A100. The double cover 2.J2 occurs as a subgroup of the Conway group Co0.
J2 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
w2 + w + 1 = 0, then J2 is generated by the two matrices
and
These matrices satisfy the equations
J2 is thus a Hurwitz group, a finite homomorphic image of the triangle group.
The matrix representation given above constitutes an embedding into Dickson's group G2. There is only one conjugacy class of J2 in G2. Every subgroup J2 contained in G2 extends to a subgroup J2:2=Aut in G2:2=Aut. G2 is in turn isomorphic to a subgroup of the Conway group Co1.

Maximal subgroups

There are 9 conjugacy classes of maximal subgroups of J2. Some are here described in terms of action on the Hall–Janko graph.
The maximum order of any element is 15. As permutations, elements act on the 100 vertices of the Hall–Janko graph.
OrderNo. elementsCycle structure and conjugacy
1 = 11 = 11 class
2 = 2315 = 32 · 5 · 7240, 1 class
2 = 22520 = 23 · 32 · 5 · 7250, 1 class
3 = 3560 = 24 · 5 · 7330, 1 class
3 = 316800 = 25 · 3 · 52 · 7332, 1 class
4 = 226300 = 22 · 32 · 52 · 726420, 1 class
5 = 54032 = 26 · 32 · 7520, 2 classes, power equivalent
5 = 524192 = 27 · 33 · 7520, 2 classes, power equivalent
6 = 2 · 325200 = 24 · 32 · 52 · 72436612, 1 class
6 = 2 · 350400 = 25 · 32 · 52 · 722616, 1 class
7 = 786400 = 27 · 33 · 52714, 1 class
8 = 2375600 = 24 · 33 · 52 · 72343810, 1 class
10 = 2 · 560480 = 26 · 33 · 5 · 71010, 2 classes, power equivalent
10 = 2 · 5120960 = 27 · 33 · 5 · 754108, 2 classes, power equivalent
12 = 22 · 350400 = 25 · 32 · 52 · 7324262126, 1 class
15 = 3 · 580640 = 28 · 32 · 5 · 752156, 2 classes, power equivalent