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In this paper, we consider all the Janko sporadic groups J1, J2, J3 and J4 (with orders 175560, 604800, 50232960 and 86775571046077562880, respectively) with a new concept called the markaracter- and Q-conjugacy character tables, which enables us to discuss marks and characters for a finite group on a common basis of Q-conjugacy relationships between their cyclic subgroups. Then by using GAP (Groups, Algorithms and Programming) package we calculate all their dominant classes enabling us to find all possible Q-conjugacy characters for these sporadic groups. Finally, we prove in a main theorem that all twenty six simple sporadic groups are unmatured. [ABSTRACT FROM AUTHOR]

For q ∈ {7, 8, 9, 11, 13, 16}, we consider the primitive actions of L2(q) and use Key-Moori Method 1 as described in [Codes, designs and graphs from the Janko groups J1 and J2, J. Combin. Math. Combin. Comput., 40 (2002) 143-159., Correction to: "Codes, designs and graphs from the Janko groups J1 and J2" [J. Combin. Math. Combin. Comput. 40 (2002) 143-159], J. Combin. Math. Combin. Comput., 64 (2008) 153.] to construct designs from the orbits of the point stabilisers and from any union of these orbits. We also use Key-Moori Method 2 (see Information security, coding theory and related combinatorics, NATO Sci. Peace Secur. Ser. D Inf. Commun. Secur., IOS Amsterdam, 29 (2011) 202-230.) to determine the designs from the maximal subgroups and the conjugacy classes of elements of these groups. The incidence matrices of these designs are then used to generate associated binary codes. The full automorphisms of these designs and codes are also determined. [ABSTRACT FROM AUTHOR]

If G is a finite group and X a conjugacy class of elements of G, then we define rank(G : X) to be the minimum number of elements of X generating G. In the present paper we study the rank(Ji : X) for all conjugacy classes of Ji, where i = 1; 2; 3; 4. [ABSTRACT FROM AUTHOR]

In this paper we give a self-contained existence and uniqueness proof for the sporadic O'Nan group ON by showing that it is uniquely determined up to isomorphism by the centralizer H of a 2-central involution z. We establish for such a simple group G a presentation in terms of generators and defining relations and a faithful permutation representation of degree 2.624.832 with a uniquely determined stabilizer isomorphic to the small sporadic Janko group J1. We also calculate its character table by new methods and determine a system of representatives of the conjugacy classes of G. [ABSTRACT FROM AUTHOR]