Janko group J₃

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.

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

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

Finkelstein & Rudvalis (1974) 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

• Finkelstein, L.; Rudvalis, A. (1974), "The maximal subgroups of Janko's simple group of order 50,232,960", Journal of Algebra, 30: 122–143, doi:10.1016/0021-8693(74)90196-3, ISSN 0021-8693, MR 0354846
• R. L. Griess, Jr., The Friendly Giant, Inventiones Mathematicae 69 (1982), 1-102. p. 93: proof that J₃ is a pariah.
• Higman, Graham; McKay, John (1969), "On Janko's simple group of order 50,232,960", Bull. London Math. Soc., 1: 89–94, correction p. 219, doi:10.1112/blms/1.1.89, MR 0246955
• 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)

• MathWorld: Janko Groups
• Atlas of Finite Group Representations: J₃ version 2
• Atlas of Finite Group Representations: J₃ version 3