Janko group J4
- 221 · 33 · 5 · 7 · 113 · 23 · 29 · 31 · 37 · 43 (= 86775571046077562880)
Existence and uniqueness 
Janko found J₄ by studying groups with an involution centralizer of the form 21 + 12.3.(M22:2). Its existence and uniqueness was shown using computer calculations by Simon P. Norton and others in 1980. It has a modular representation of dimension 112 over the finite field of two elements and is the stabilizer of a certain 4995 dimensional subspace of the exterior square, a fact which Norton used to construct it, and which is the easiest way to deal with it computationally. Aschbacher & Segev (1991) and Ivanov (1992) gave computer-free proofs of uniqueness. Ivanov & Meierfrankenfeld (1999) and Ivanov (2004) gave a computer-free proof of existence by constructing it as an amalgams of groups 210:SL5(2) and (210:24:A8):2 over a group 210:24:A8.
The smallest faithful complex representation has dimension 1333; there are two complex conjugate representations of this dimension. The smallest faithful representation over any field is a 112 dimensional representation over the field of 2 elements.
The smallest permutation representation is on 173067389 points, with point stabilizer of the form 211M24. These points can be identified with certain "special vectors" in the 112 dimensional representation.
It has a presentation in terms of three generators a, b, and c as
Maximal subgroups 
Kleidman & Wilson (1988) showed that J4 has 13 conjugacy classes of maximal subgroups.
- 211:M24 - containing Sylow 2-subgroups and Sylow 3-subgroups; also containing 211:(M22:2), centralizer of involution of class 2B
- 21+12.3.(M22:2) - centralizer of involution of class 2A - containing Sylow 2-subgroups and Sylow 3-subgroups
- 23+12.(S5 × PSL(3,2)) - containing Sylow 2-subgroups
- 111+2:(5 × GL(2,3)) - normalizer of Sylow 11-subgroup
- U3(3) - containing Sylow 3-subgroups
- 29:28 Frobenius group
- 43:14 Frobenius group
- 37:12 Frobenius group
A Sylow 3-subgroup is a Heisenberg group: order 27, non-abelian, all non-trivial elements of order 3
- Aschbacher, Michael; Segev, Yoav (1991), "The uniqueness of groups of type J₄", Inventiones Mathematicae 105 (3): 589–607, doi:10.1007/BF01232280, ISSN 0020-9910, MR1117152
- D.J. Benson The simple group J4, PhD Thesis, Cambridge 1981, http://www.maths.abdn.ac.uk/~bensondj/papers/b/benson/the-simple-group-J4.pdf
- Ivanov, A. A. (1992), "A presentation for J₄", Proceedings of the London Mathematical Society. Third Series 64 (2): 369–396, doi:10.1112/plms/s3-64.2.369, ISSN 0024-6115, MR1143229
- Ivanov, A. A.; Meierfrankenfeld, Ulrich (1999), "A computer-free construction of J₄", Journal of Algebra 219 (1): 113–172, doi:10.1006/jabr.1999.7851, ISSN 0021-8693, MR1707666
- Ivanov, A. A. The fourth Janko group. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, Oxford, 2004. xvi+233 pp. ISBN 0-19-852759-4 MR2124803
- Z. Janko, A new finite simple group of order 86,775,570,046,077,562,880 which possesses M24 and the full covering group of M22 as subgroups, J. Algebra 42 (1976) 564-596.doi:10.1016/0021-8693(76)90115-0 (The title of this paper is incorrect, as the full covering group of M22 was later discovered to be larger: center of order 12, not 6.)
- Kleidman, Peter B.; Wilson, Robert A. (1988), "The maximal subgroups of J4", Proceedings of the London Mathematical Society. Third Series 56 (3): 484–510, doi:10.1112/plms/s3-56.3.484, ISSN 0024-6115, MR 931511
- S. P. Norton The construction of J4 in The Santa Cruz conference on finite groups (Ed. Cooperstein, Mason) Amer. Math. Soc 1980.