The Spectrum of G- $${ { OD}}_{\lambda }(3,4,v)$$ O D λ ( 3 , 4 , v )

Abstract

An ordered design $${ OD}_{\lambda }(t,k,v)$$ is a k-X array A, where $$|X|=v$$ , satisfying that each row of A contains k distinct elements of X and every t columns of A contain each ordered t-subset of X exactly $$\lambda $$ times. The subgroup of conjugation under which a k-X array A is invariant is called the conjugate invariant subgroup of A. If an $${ OD}_{\lambda }(t,k,v)$$ has a conjugate invariant subgroup G, then it is denoted by G- $${ OD}_{\lambda }(t,k,v)$$ . In this paper, we give the existence of G- $${ OD}_{\lambda }(3,4,v)~(\lambda \ge 2)$$ for $$C_{3}~ (\langle (123)(4)\rangle ), S_{3}$$ (symmetric group of degree 3), $$C_{4} ~(\langle (1234)\rangle )$$ and $$K_{4}$$ (Klein 4-group). Together with the known results, we almost completely determine the spectrum of G- $${ OD}_{\lambda }(3,4,v)$$ for G a subgroup of $$S_{4}$$ .