List of small groups

The following list contains a selection of finite groups of small order.

This list can be used to find out to which known finite groups, a group G is isomorphic. The first one determines the order of G and compares it to the listed groups of the same order. Is known if G is abelian ( commutative ), so you can exclude some groups. Then comparing the order of individual elements of G with the elements of the listed groups, which can be determined uniquely G up to isomorphism.

Glossary

  • Is the cyclic group of order n ( which is also written as ).
  • Is the dihedral group of order 2n
  • Is the symmetric group of degree n, with n! Permutations of n elements.
  • : The alternating group of degree n, with n / 2 permutations of n elements! .
  • : Is the group of order 4n dizyklische.
  • Is the Klein four- group of order 4
  • Is the quaternion group of order for.

The notation used to denote the direct product of the groups G and H. It is noted, if a group is abelian or simple. ( For groups of order n <60 are the simple groups exactly the cyclic groups, with n from the set of prime numbers. ) In the Cycles graph of groups is the neutral element is represented by a filled black circle. Order 16 is the smallest order for which the group structure is not uniquely determined by the Cycles graphs: The non- abelian modular group and have the same Cycles graph and the same ( modular ) sub-group association, but are not isomorphic.

It should be noted that means, that there are 3 sub-groups of the type ( not by the coset ).

List of non abelian groups up to order 16

List of all groups to order 16

"Small groups library"

The computer algebra system GAP contains the program library Small Groups library, which contains a description of groups of small order. These are all listed up to isomorphism. Currently, the library contains groups following order:

  • All the order to 2000, except for the 49,487,365,422 groups of order 1024 ( stay 423 164 062 groups);
  • All of order 55 and 74 (92 groups);
  • All of order qn × p with qn divides 28, 36, 55 or 74, and p is any prime different from q;
  • All groups whose order n in at most three primes is dismantled.

This library was created by Hans Ulrich Besche, Bettina Eick and Eamonn O'Brien.

Klein four-group Classification of finite simple groups Lagrange's theorem (group theory) Semidirect product Quasidihedral group Trivial group GAP (computer algebra system) Dihedral group of order 6 Tetrahedral symmetry
515472
de