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

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.
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 [Modular Modular representation theory|representation theory|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.

Constructions

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:
and

Using the subgroup PSL(2,16)

The automorphism group J3:2 can be constructed by starting with the subgroup PSL: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:
where is the Frobenius automorphism of order 4, and is the unique 17-cycle that sends
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 presentation for J3 in terms of generators a, b, c, d is

Maximal subgroups

found the 9 conjugacy classes of maximal subgroups of J3 as follows:
No.StructureOrderIndexComments
1L2:28,160
= 25·3·5·17		6,156
= 22·34·19
2,3L23,420
= 22·32·5·19		14,688
= 25·33·17		two classes, fused by an outer automorphism
424: 2,880
= 26·32·5		17,442
= 2·33·17·19
5L22,448
= 24·32·17		20,520
= 23·33·5·19		centralizer of an outer automorphism of order 2
6:222,160
= 24·33·5		23,256
= 23·32·17·19		normalizer of a subgroup of order 3
732+1+2:81,944
= 23·35		25,840
= 24·5·17·19		normalizer of a Sylow 3-subgroup
82:A51,920
= 27·3·5		26,163
= 34·17·19		centralizer of involution
922+4: 1,152
= 27·32		43,605
= 33·5·17·19