Janko group J3

In the area of modern algebra known as group theory, the Janko group J3 or the Higman-Janko-McKay group HJM is a sporadic simple group of order
 * 27·35·5·17·19 = 50232960.

History and properties
J3 is one of the 26 Sporadic groups and was predicted by Zvonimir Janko in 1969 as one of two new simple groups having 21+4:A5 as a centralizer of an involution (the other is the Janko group J2). J3 was shown to exist by.

In 1982 R. L. Griess showed that J3 cannot be a subquotient of the monster group. Thus it is one of the 6 sporadic groups called the pariahs.

J3 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. 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.

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 J3:2 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 J3:2. One then defines the following relation:

$$\left(\begin{matrix}1&1\\1&0\end{matrix}\sigma t_{(\nu,\nu7)}\right)^5=1$$

where $$\sigma$$ is the Frobenius automorphism or order 4, and $$t_{(\nu,\nu7)}$$ is the unique 17-cycle that sends

$$\infty\rightarrow0\rightarrow1\rightarrow7$$

Curtis showed, using a computer, that this relation is sufficient to define J3:2.

Using a presentation
In terms of generators a, b, c, and d its automorphism group J3:2 can be presented as $$a^{17} = b^8 = a^ba^{-2} = c^2 = b^cb^3 = (abc)^4 = (ac)^{17} = d^2 = [d, a] = [d, b] = (a^3b^{-3}cd)^5 = 1.$$

A presentation for J3 in terms of (different) generators a, b, c, d is $$a^{19} = b^9 = a^ba^2 = c^2 = d^2 = (bc)^2 = (bd)^2 = (ac)^3 = (ad)^3 = (a^2ca^{-3}d)^3 = 1.$$

Maximal subgroups
found the 9 conjugacy classes of maximal subgroups of J3 as follows:


 * PSL(2,16):2, order 8160
 * PSL(2,19), order 3420
 * PSL(2,19), conjugate to preceding class in J3:2
 * 24: (3 × A5), order 2880
 * PSL(2,17), order 2448
 * (3 × A6):22, order 2160 - normalizer of subgroup of order 3
 * 32+1+2:8, order 1944 - normalizer of Sylow 3-subgroup
 * 21+4:A5, order 1920 - centralizer of involution
 * 22+4: (3 × S3), order 1152