The List of small groups reference article from the English Wikipedia on 24-Jul-2004 (provided by Fixed Reference: snapshots of Wikipedia from wikipedia.org)

List of small groups

In mathematics, the finite groups of small order can be listed up to group isomorphism.

Glossary

• C_n: the cyclic group of order n
• D_n: the dihedral group of order n
• S_n: the symmetric group of degree n, containing the n permutations of n elements.
• A_n: the alternating group of degree n, containing the n!/2 even permutations of n elements.

The notation G × H stands for the direct product of the two groups. Abelian and simple groups are noted. (For groups of order n < 60, the simple groups are precisely the cyclic groups C_n, where n is prime.)

List

Order	Group	Properties
1	trivial group = C1 = S1 = A2	abelian
2	C2 = S2	abelian, simple, the smallest non-trivial group
3	C3 = A3	abelian, simple
4	C4	abelian,
	Klein four-group = C2 × C2 = D4	abelian, the smallest non-cyclic group
5	C5	abelian, simple
6	C6 = C2 × C3	abelian
	S3 = D6	the smallest non-abelian group
7	C7	abelian, simple
8	C8	abelian
	C2 × C4	abelian
	C2 × C2 × C2	abelian
	D8	non-abelian
	Quaternion group = Q8	the smallest non-abelian group, each of whose subgroups is normal
9	C9	abelian
	C3 × C3	abelian
10	C10 = C2 × C5	abelian
	D10	non-abelian
11	C11	abelian, simple
12	C12 = C4 × C3	abelian
	C2 × C6 = C2 × C2 × C3	abelian
	D12	non-abelian
	A4	non-abelian
	the semidirect product of C3 and C4, where C4 acts on C3 by inversion	non-abelian
13	C13	abelian, simple
14	C14 = C2 × C7	abelian
	D14	non-abelian
15	C15 = C3 × C5	abelian

Please add higher orders, and/or more information about the groups (maximal subgroups, normal subgroups, character tables etc.)

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The group theoretical computer algebra system GAP (available for free at http://www.gap-system.org/ ) contains the "Small Groups library": it provides access to descriptions of the groups of "small" order. The groups are listed up to isomorphism. At present, the library contains the following groups:

• those of order at most 2000 except 1024 (423 164 062 groups);
• those of order 5^5 and 7^4 (92 groups);
• those of order q^n * p where q^n divides 2^8, 3^6, 5^5 or 7^4 and p is an arbitrary prime which differs from q;
• those whose order factorises into at most 3 primes.

It contains explicit descriptions of the available groups in computer readable format.

The library has been constructed and prepared by Hans Ulrich Besche, Bettina Eick and Eamonn O'Brien; see http://www.tu-bs.de/~hubesche/small.html .

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