List of small order groups

Each group in the list is indicated by its index in the library of small groups as G _o ⁱ , where o is the order of the group, and i is its index among groups of this order.

Common group names are also used:

• Z _n is a cyclic group of order n (the notation C _{n is} also used. The group is isomorphic to the additive group Z / n Z ).
  • K ₄ is the fourth Klein group of order, the same as Z ₂ × Z ₂ or Dih ₂ .
• Dih _n is a dihedral group of order 2 n (the notation D _n or D _{2 n is} often used)
• S _n is a symmetric group of order n containing n ! permutations of n elements.
• A _n is an alternating group of degree n containing n ! / 2 even permutations of n elements.
• Dic _n or Q _4n is a dicyclic group of order 4 n .
  • Q ₈ is a group of quaternions of order 8, also Dic ₂ .

The notation Z _n and Dih _{n is} preferable, since there are notation C _n and D _n for point groups in three-dimensional space.

The designation G × H is used to directly product two groups. G ⁿ denotes the direct product of the group itself by itself n times. G ⋊ H denotes a semidirect product , where H acts on G.

Abelian and simple groups are listed. (For groups of order n <60, simple groups are exactly cyclic groups Z _n for primes n .) The equal sign ("=") means an isomorphism.

The neutral element in the cycle graph is represented by a black circle. The cycle graph defines a group uniquely only for groups whose order is less than 16.

In the lists of subgroups, the trivial group and the group itself are not listed. If there are several isomorphic subgroups, their number is indicated in brackets.

Finite Abelian groups are either cyclic groups or their direct product, see the article Abelian group .

Groups with a small order equal to the power of a prime p ⁿ :

• Order p : all such groups are cyclic.
• Order p ² : there are two groups, both Abelian.
• Order p ³ : 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 p ² and a cyclic group of order p . Another group is the quaternion group for p = 2 and the Heisenberg group modulo p for p '> 2.
• Order p ⁴ : the classification of groups is complicated and becomes more complicated with increasing p .

Most small-order groups have a Sylow p -subgroup P with normal p- complement N for some prime p dividing the order, so that they can be classified in terms of possible primes of p , p -groups P , groups N and actions P on N. In a sense, this reduces the classification of such groups to the classification of p -groups. Small-order groups that do not have normal p- complement include:

• Order 24: Symmetric Group S ₄
• Order 48: binary octahedral group and product S ₄ × Z / 2 Z
• Order 60: alternating group A ₅ .

The GAP computer algebra system contains a “Small Group Library” that provides descriptions of small-order groups. Groups are listed up to isomorphism . The library currently contains the following groups: ^[59]

• groups whose order does not exceed 2000, with the exception of order 1024 (423 164 062 groups in the library. Groups of order 1024 are skipped because there are 49 487 365 422 non - isomorphic 2-groups of order 1024.);
• groups whose order is not divided by a cube, with the order of up to 50,000 (395,703 groups);
• groups whose order is not divisible by square;
• groups of order p ⁿ for n at most 6 and prime p ;
• groups of order p ⁷ for p = 3, 5, 7, 11 (907,489 groups);
• groups of order q ⁿ × p , where q ⁿ divides 2 ⁸ , 3 ⁶ , 5 ⁵ or 7 ⁴ and p is an arbitrary prime number other than q ;
• groups whose order is the product of no more than three primes.

• Classification of simple finite groups

• HSM Coxeter, WOJ Moser. Generators and Relations for Discrete Groups. - New York: Springer-Verlag, 1980 .-- ISBN 0-387-09212-9 . , Table 1, Non-Abelian groups of order <32.
• Marshall Hall Jr., James K. Senior. The Groups of Order 2 ⁿ ( n ≤ 6 ). - Macmillan, 1964.

• Particular groups in the Group Properties Wiki
• HU Besche, B. Eick, E. O'Brien. small group library (unopened) (inaccessible link) . Archived on March 5, 2012.
• RJ Mathar. Plots of cycle graphs of the finite groups up to order 36 (neopr.) (2014).