Database of Finite Perfect Groups

1. Introduction:

This database was intended to contain all finite perfect groups of order up to one million. In fact, due to the very large number of groups involved, it is incomplete in certain places (for details, see Section 3). It is totally complete up to order 50000. There is one database file for each group, which defines the group by means of a finite presentation. Permutation representations may also be obtained. The groups are organized in the same way as in the corresponding tables in Chapter 5.3 of the book "Perfect Groups" by D.F. Holt and W. Plesken, Oxford University Press, 1989. The notation used for the names of the files presented considerable difficulties, but it follows that used in the book as far as possible.

2. Contents of the individual files:

Within each file, the group is always named gp, and is defined by means of a finite presentation. Usually, the names of the generators are the same as those used in "Perfect Groups", but we have replaced capital letters like S by s1 for historical reasons (Cayley did not distinguish between lower and upper case letters). In most cases, many more generators and relations are given than are strictly necessary. This is because most of the groups consist of an extension of a p-group N by a simple or quasisimple group, and the presentation includes a complete power-commutator presentation of N.

After the definition of the group, one or more sequences of subgroups are defined. These are intended for the construction of faithful permutation representations. For further details, see Permutation Representations for Database of Finite Perfect Groups.

3. Contents of the database:

As in the book "Perfect Groups", the groups are arranged in classes. These are as follows.

• Class A5#2 (complete up to order 50000)
• Classes A5#3 and A5C2#3 (complete up to order 1000000).
• Classes A5#5 and A5C2#5 (complete up to order 1000000).
• Classes A5#7 and A5C2#7 (complete up to order 1000000).
• Classes A5#p and A5C2#p for p > 11 (complete up to order 1000000).
• Classes A5 2^4' # 3 etc. (complete up to order 1000000).
• Class L32#2 (complete up to order 50000)
• Classes L32#3 and L32C2#3 (complete up to order 1000000).
• Classes L32#7 and L32C2#7 (complete up to order 1000000).
• Classes L32#11 and L32C2#11 (complete up to order 1000000).
• Classes A6#2 and A6C3#2 (complete up to order 200000, 50000, resp.).
• Classes A6#3 and A6C2#3 (complete up to order 1000000).
• Class L28#2 (complete up to order 1000000).
• Class L211#2 (complete up to order 1000000).
• Classes L211#3 and L211C2#3 (complete up to order 1000000).
• Classes L211#11 and L211C2#11 (complete up to order 1000000).
• Classes L213#13 and L213C2#13 (complete up to order 1000000).
• Class L217#2 (complete up to order 1000000).
• Other groups L2q and L2qC2 (complete up to order 1000000).
• Classes A7#2 and A7C3#2 (complete up to order 1000000).
• Class L33#3 (complete up to order 1000000).
• Class U33#2 (complete up to order 1000000).
• Class A8#2 (complete up to order 1000000).
• Classes L34#2 and L34C3#2 (complete up to order 1000000).
• Other simple and quasisimple groups (complete up to order 1000000).
• Class A5xA5#2 (complete up to order 50000, but includes two larger groups).
• Direct products of simple groups and their covers (complete up to order 50000).

For further details on the notation, see Section 4, but for now note simply that A5#2 means extensions of 2-group by A5, and A5C2 means the 2-fold covering group of A5 (i.e. SL(2,5)). For the complete list of all of the database files arranged according to these classes, see Contents of Database of Finite Perfect Groups.

In fact, the database of perfect groups is complete up to order one million, with the following exceptions. It is complete everywhere up to order 50000.

• Classes A5#2 and L32#2 are complete only up to order 50000.
• Class A6#2 is complete only up to order 200000.
• Class A6C3#2 is complete only up to order 50000.
• Extensions of groups P x Q by H = A5, A5C2, L32, L32C2, A6 or A6C2, where P and Q are groups of prime power orders p and q for distinct primes p and q, and H acts nontrivially on both P and Q, are not included. These all have orders bigger than 50000.
• Extensions of nilpotent groups by direct products of simple groups are complete only up to order 50000. However, the only two interesting cases, in which A5 x A5 acts nontrivially on a module of order 2^8, are included.

4. Notation for the groups:

The names for simple and quasisimple groups are fairly natural. So, we have A5 for the alternating group of degree 5 and A5C2 for its 2-fold covering group, which is of course isomorphic to SL(2,5).

A typical group in the database will be in a class like A5#2 - the noncentral extensions of 2-groups by A5. Unfortunately, it is syntactically inconvenient to use # as a separator in a Magma database file name, so we have used _. We then have names like A5_2_53, which means the third group of order 2^5.60 in the class A5#2.

Occasionally, some common-sense is required to resolve potential ambiguities. For example, A5_2_911 means the 11th group of order 2^9.60 in the class A5#2, whereas L211_2_102 means the second group of order 2^10.660 in the class L2(11)#2.

For direct products, we have obvious names like A5xL32 and A5C2xL32. For the amalgamated central product, we have used A5xL32q2.

Subscreens

• Contents of Database of Finite Perfect Groups
• Permutation Representations for Database of Finite Perfect Groups