# Janko group J3

In mathematics, the third Janko group J3, also known as the Higman-Janko-McKay group, is a finite simple sporadic group of order 50232960 = 27 · 35 · 5 · 17 · 19. Evidence for its existence was uncovered by Zvonimir Janko (1969) and it was shown to exist by Graham Higman and John McKay (1969). Janko predicted both J3 and J2 as simple groups having 21+4:A5 as a centralizer of an involution.

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 field with 4 elements. Weiss (1982) constructed it via an underlying geometry. and it has a modular representation of dimension eighteen over the finite field of nine elements.

J3 is one of the 6 sporadic simple groups called the pariahs, because (Griess 1982) showed that it is not found within the Monster group.

## Presentations

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

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

## Maximal subgroups

Finkelstein & Rudvalis (1974) showed that J3 has 9 conjugacy classes of maximal subgroups:

• 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

## References

• 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.MR 0244371
• Richard Weiss, "A Geometric Construction of Janko's Group J3", Math. Zeitschrift 179 pp 91-95 (1982)