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# Finite Groups and Designs

DOI link for Finite Groups and Designs

Finite Groups and Designs book

# Finite Groups and Designs

DOI link for Finite Groups and Designs

Finite Groups and Designs book

## ABSTRACT

Table 9.62 has eight columns. The first column is an index of the form d.i, where d is the degree of the primitive permutation group and i is a serial index distinguishing primitive groups of the same degree. Column 2 contains the name of the group, which often also provides structural information about the group. Column 3 contains the order of the group. Column 4 provides information about the depth of transitivity of the group. For example, 3p! indicates that the group is sharply 3-primitive. The “p” notation indicates primitivity, the numeral 3 indicates the depth (the stabilizer of a point is 2-primitive on d − 1 points), and the exclamation mark indicates that the stabilizer of two points is sharply primitive (hence, also regular) on d− 2 points. A notation such as 3t! means that the group is sharply 3-transitive. Similarly, 3t indicates that the group is 3-transitive, 3* that the group is 3-homogeneous, while 3*! indicates that the group is sharply 3-homogeneous. If the group is 1-primitive but

not 2-primitive, then the stabilizer of a point is not primitive on d− 1 points, and in these cases the orbit lengths for the action of the point-stabilizer are provided. For example, group 15.1 (A6 acting on 15 points) is just primitive but not 2-primitive. The stabilizer of a point, which happens to be isomorphic to S4, has 3 orbits on points, of lengths 1, 6, and 8. A superscript over an orbit length indicates the number of times that orbit length is encountered, so, for example, for group 16.1 column 4 contains [1, 53], which means that the stabilizer of a point has 3 orbits of length 5, besides the fixed point.