Search Results for “list of small groups”

Source: List of small groups

The following list in mathematics contains the finite groups of small order up to group isomorphism.

Counts

groups

Glossary

show the presentation of a group.

List of small abelian groups

The finite abelian groups are either cyclic groups, or direct products thereof; see Abelian group. The numbers of nonisomorphic abelian groups of orders n = 1, 2, ... are

1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 5, 1, 2, 1, 2, ... (sequence A000688 in the OEIS)
For labeled abelian groups, see OEIS: A034382.

List of small non-abelian groups

The numbers of non-abelian groups, by order, are counted by (sequence A060689 in the OEIS). However, many orders have no non-abelian groups. The orders for which a non-abelian group exists are

6, 8, 10, 12, 14, 16, 18, 20, 21, 22, 24, 26, 27, 28, 30, 32, 34, 36, 38, 39, 40, 42, 44, 46, 48, 50, ... (sequence A060652 in the OEIS)

Classifying groups of small order

Small groups of prime power order pn are given as follows:

Order p: The only group is cyclic.
Order p2: There are just two groups, both abelian.
Order p3: There are three abelian groups, and two non-abelian groups. One of the non-abelian groups is the semidirect product of a normal cyclic subgroup of order p2 by a cyclic group of order p. The other is the quaternion group for p = 2 and a group of exponent p for p > 2.
Order p4: The classification is complicated, and gets much harder as the exponent of p increases.
Most groups of small order have a Sylow p subgroup P with a normal p-complement N for some prime p dividing the order, so can be classified in terms of the possible primes p, p-groups P, groups N, and actions of P on N. In some sense this reduces the classification of these groups to the classification of p-groups. Some of the small groups that do not have a normal p-complement include:

Order 24: The symmetric group S4
Order 48: The binary octahedral group and the product S4 × Z2
Order 60: The alternating group A5.
The smallest order for which it is not known how many nonisomorphic groups there are is 2048 = 211.

Small Groups Library

The GAP computer algebra system contains a package called the "Small Groups library," which provides access to descriptions of small order groups. The groups are listed up to isomorphism. At present, the library contains the following groups:

those of order at most 2000 except for order 1024 (423164062 groups in the library; the ones of order 1024 had to be skipped, as there are additional 49487367289 nonisomorphic 2-groups of order 1024);
those of cubefree order at most 50000 (395 703 groups);
those of squarefree order;
those of order pn for n at most 6 and p prime;
those of order p7 for p = 3, 5, 7, 11 (907 489 groups);
those of order pqn where qn divides 28, 36, 55 or 74 and p is an arbitrary prime which differs from q;
those whose orders factorise into at most 3 primes (not necessarily distinct).
It contains explicit descriptions of the available groups in computer readable format.
The smallest order for which the Small Groups library does not have information is 1024.

Notes

References

Coxeter, H. S. M. & Moser, W. O. J. (1980). Generators and Relations for Discrete Groups. New York: Springer-Verlag. ISBN 0-387-09212-9., Table 1, Nonabelian groups order<32.
Hall, Jr., Marshall; Senior, James K. (1964). "The Groups of Order 2n (n ≤ 6)". MathSciNet. Macmillan. MR 0168631. A catalog of the 340 groups of order dividing 64 with tables of defining relations, constants, and lattice of subgroups of each group.

External links

Particular groups in the Group Properties Wiki
Besche, H.U.; Eick, B.; O'Brien, E. "Small Group Library". Archived from the original on 2012-03-05.
GroupNames database
Hall, Jr., Marshall; Senior, James Kuhn (1964). The Groups of Order 2n (n ≤ 6). New York: Macmillan / London: Collier-Macmillan Ltd. LCCN 64016861