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 orderHistory 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 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.Presentations
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 isfound the 9 conjugacy classes of maximal subgroups of J3 as follows:
- PSL:2, order 8160
- PSL, order 3420
- PSL, conjugate to preceding class in J3:2
- 24:, order 2880
- PSL, order 2448
- :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:, order 1152
