Janko group J2
Algebraic structure → Group theory Group theory 


Modular groups

Infinite dimensional Lie group

In the area of modern algebra known as group theory, the Janko group J_{2} or the HallJanko group HJ is a sporadic simple group of order
 2^{7} · 3^{3} · 5^{2} · 7 = 604800.
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 Hall and Wales (1968) as a rank 3 permutation group on 100 points.
Both the Schur multiplier and the outer automorphism group have order 2.
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,^{[1]} 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
and
These matrices satisfy the equations
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 are two conjugacy classes of HJ in G_{2}(4), and they are equivalent under the automorphism on the field F_{4}. Their intersection (the "real" subgroup) is simple of order 6048. 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.
 U_{3}(3) order 6048 – onepoint 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 168subgroups, thus fixes them under conjugacy. Its centralizer has structure 4.S_{4}, which contains 6 additional involutions.
 3.PGL(2,9) order 2160 – has a subquotient A_{6}
 2^{1+4}:A_{5} order 1920 – centralizer of involution moving 80 points
 2^{2+4}:(3 × S_{3}) order 1152
 A_{4} × A_{5} order 720
 Containing 2^{2} × A_{5} (order 240), centralizer of 3 involutions each moving 100 points
 A_{5} × D_{10} order 600
 PGL(2,7) order 336
 5^{2}:D_{12} order 300
 A_{5} 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 = 3^{2} · 5 · 7  2^{40}, 1 class 
2520 = 2^{3} · 3^{2} · 5 · 7  2^{50}, 1 class  
3 = 3  560 = 2^{4} · 5 · 7  3^{30}, 1 class 
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 
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 
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 
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 
References
 Robert L. Griess, Jr., "Twelve Sporadic Groups", SpringerVerlag, 1998.
 Hall, Marshall; Wales, David (1968), "The simple group of order 604,800", Journal of Algebra, 9: 417–450, doi:10.1016/00218693(68)900148, ISSN 00218693, MR 0240192 (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, MR 0244371
 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:10.1016/00218693(69)901136, MR0251133, ISSN 00218693