The transitive groups of degree 48 and some applications

Abstract

The primary purpose of this paper is to report on the successful enumeration in Magma of representatives of the $195826352$ conjugacy classes of transitive subgroups of the symmetric group $S_{48}$ of degree 48. In addition, we have determined that 25707 of these groups are minimal transitive and that 713 of them are elusive. The minimal transitive examples have been used to enumerate the vertex-transitive groups of degree 48, of which there are $1538868366$, all but 0.1625% of which arise as Cayley graphs. We have also found that the largest number of elements required to generate any of these groups is 10, and we have used this fact to improve previous general bounds of the third author on the number of elements required to generate an arbitrary transitive permutation group of a given degree. The details of the proof of this improved bound will be published as a separate paper.

Introduction

Since late in the 19th century, significant effort has been devoted to compiling catalogues and databases of various types of groups, including complete lists of (representatives of the conjugacy classes of) the transitive and primitive subgroups of the symmetric groups of small degree. For the transitive groups, earlier references include [19], [18] (with corrections in [22]) for degrees up to 12, [10] for degrees up to 31, [3] for the significantly more difficult case of degree 32, and [9] for degrees 33–47. (There are $2801324$ groups of degree 32, and $501045$ groups of all other degrees up to 47.) Apart from the early work of Miller, these lists have been compiled by computer, using GAP [7] and Magma [2].

Degree 48 is once again significantly more difficult than earlier degrees, because there are many more groups and the computations involved need more time and computer memory. The main purpose of this paper is to report on the successful enumeration of conjugacy class representatives of the transitive subgroups of degree 48, which is the topic of Section 2. The result is summarised in the following theorem.

Theorem 1.1

There are $195826352$ conjugacy classes of transitive subgroups of the symmetric group $S_{48}$.

The complete list of these subgroups is available in Magma from an optional database that can be downloaded by users from the Magma website.

The computations were carried out in Magma and the final successful runs required about one year of cpu-time (this does not include unsuccessful attempts that had to be aborted for various reasons). The methods used are theoretically essentially the same as those described in [3] and [9] for degrees $32-48$, but completing the calculations in degree 48 proved to be significantly more challenging, and parts of the Magma code needed to be optimised or sometimes re-designed to cope with the larger amounts of data. For this reason they took more than two years of real time to complete.

Although the computations were carried out serially on a single processor, they are intrinsically extremely parallelisable: about 98% of the cpu-time was for imprimitive groups with blocks of size 2 and, as we shall explain shortly, that case splits into 25000 independent calculations.

We anticipate that it would be feasible to extend the catalogues up to degree 63, but that degree 64 will remain out of range for the foreseeable future.

We subsequently used our catalogue to identify those groups of degree 48 that are minimal transitive (that is, they have no proper transitive subgroups) and those that are elusive (that is, they contain no fixed-point-free elements of prime order). We shall report on this in Section 3. The various counts of groups involved are summarised in Table 1, where the imprimitive groups have been counted according to the smallest size of a block of imprimitivity.

In Section 4, we describe the computation of the vertex-transitive graphs of order 48, along with some associated data.

We denote the smallest size of a generating set of a group G by $d(G)$. We have established that $d(G)⩽10$ for all transitive groups of degree 48. In fact, the only examples with $d(G)=10$ have minimal block size 3 and have 10-generator transitive groups of degree 32 as quotients. The groups with block size 2 all satisfy $d(G)⩽9$.

We shall discuss how we computed this information at the beginning of Section 5. There we shall explain how these bounds on $d(G)$ have enabled the third author to remove the exceptional cases of the general bound on $d(G)$ for transitive permutation groups of degree n that he established in [23], and thereby to complete the proof of the following result, where logarithms are to the base 2.

Theorem 1.2

Let G be a transitive permutation group of degree n. Then

$$d(G)⩽\lfloor \frac{cn}{\sqrt{\log n}}\rfloor$$

where $c:=\frac{\sqrt{3}}{2}$.

(It was proved by Lucchini, Menegazzo and Morigi in [15] that this result holds for some unspecified constant c.) Since the proof of this result involves some lengthy case-by-case analyses, we shall just summarise it in Section 5 of this paper, and the details will be published separately (see [24]).

In a related application, the third author is now able to bound the constant d in the result proved in [16] that $d(G)⩽d\log n/\sqrt{\log \log n}$ for primitive subgroups G of $S_n$.

The computations described in this paper were carried out on a number of different machines; the most time consuming were done using a model of type “Intel(R) Xeon(R) 3.10 GHz” with about 400 GB of RAM.

Notation

For a finite group G, we will write $\mathrm{\Phi }(G)$, $R(G)$, $[G,G]$ for the Frattini subgroup, soluble radical, and derived subgroup of G, respectively. We will mostly use the notation from [26] for group names, although we simply write n for the cyclic group of order n when there is no danger of confusion.