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.[1] 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) harvtxt error: no target: CITEREFWeiss1982 (help) 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 (different) generators a, b, c, d is
Constructions[]
J3 can be constructed by many different generators.[2] Two from the ATLAS list are 18x18 matrices over the finite field of order 9, with matrix multiplication carried out with finite field arithmetic:
Z. Janko, Some new finite simple groups of finite order, 1969 Symposia Mathematica (INDAM, Rome, 1967/68), Vol. 1 pp. 25–64 Academic Press, London, and in The theory of finite groups (Edited by Brauer and Sah) p. 63-64, Benjamin, 1969.MR0244371
Richard Weiss, "A Geometric Construction of Janko's Group J3", Math. Zeitschrift 179 pp 91–95 (1982)