Janko group J3

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 Graham Higman and John McKay (1969).

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. Weiss (1982) 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 17 = b 8 = a b a − 2 = c 2 = b c b 3 = ( a b c ) 4 = ( a c ) 17 = d 2 = [ d , a ] = [ d , b ] = ( a 3 b − 3 c d ) 5 = 1.

A presentation for J₃ in terms of (different) generators a, b, c, d is a 19 = b 9 = a b a 2 = c 2 = d 2 = ( b c ) 2 = ( b d ) 2 = ( a c ) 3 = ( a d ) 3 = ( a 2 c a − 3 d ) 3 = 1.

Maximal subgroups

Finkelstein & Rudvalis (1974) found the 9 conjugacy classes of maximal subgroups of J₃ as follows: