Column Heart A <sub> <sup> 5 </ Sub> 3 </ Sup> Sd-type Sub-orbital Quasiprimitive Permutation Group

Let G be a transitive permutation group on a setΩ. For a∈Ω, each orbit of the stabiliser subgroup Ga:={x∈G |a~x=a} onΩis called a suborbit of G, and the number of all suborbits of G is called the rank of G. Moreover. G may naturally induces an action onΩxΩ. The each orbit of this induced action is called an orbital of G. A graph defined with vertex setΩand arc set△is called an orbital graph, where△is an orbital of G. For a given permutation group, determining its orbital graphs (especially determining the length of its all suborbits, that is, the valencies of it orbital graphs) is an important research project in algebraic graph theory. For example, transitive permutation groups with a no self-conjugate suborbits are stud-ied in [7]; primitive permutation groups with small orbitals are studied in [8]; in [9], primitive permutation groups with small rank are characterized while primitive permutation groups and suborbits with small rank are determined in [20]. For more related results, see [11],[18],[19],[29]. The main purpose of this thesis is to give some properties of quasiprimitive permutation group of SD type, and determine subor-bits of those groups with socle soc(G) = A53, thus also determine valencies of G-arc transitive graphs.The main result we obtained is the following theorem.Theory 1. Let G= A53:(Out(A5) x S3) is a quasiprimitive permutation group of SD type onΩ, then G has 18 suborbits onΩ. Moreover, let ns denoted the number of all suborbits of G whose length is s, then (s, ns)= (1,1), (20,1), (30,1), (45,1), (60,1), (72,2), (120,2), (180,1), (360,8).