List of small groups

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

The list can be used to determine which known group a given finite group G is isomorphic to: first determine the order of G, then look up the candidates for that order in the list below. If you know whether G is abelian or not, some candidates can be eliminated right away. To distinguish between the remaining candidates, look at the orders of your group's elements, and match it with the orders of the candidate group's elements.

Glossary[edit]

The notations Zn and Dihn have the advantage that point groups in three dimensions Cn and Dn do not have the same notation. There are more isometry groups than these two, of the same abstract group type.

The notation G × H stands for the direct product of the two groups; Gn denotes the direct product of a group with itself n times. GH stands for a semidirect product where H acts on G; where the particular action of H on G is omitted, it is because all possible non-trivial actions result in the same product group, up to isomorphism.

Abelian and simple groups are noted. (For groups of order n < 60, the simple groups are precisely the cyclic groups Zn, for prime n.) The equality sign ("=") denotes isomorphism.

The identity element in the cycle graphs is represented by the black circle. The lowest order for which the cycle graph does not uniquely represent a group is order 16.

In the lists of subgroups, the trivial group and the group itself are not listed. Where there are multiple isomorphic subgroups, their number is indicated in parentheses.

List of small abelian groups[edit]

The finite abelian groups are either cyclic groups, or direct products thereof; see abelian groups.

Order Group Subgroups Properties Cycle graph
1 trivial group, Z1 = S1 = A2 various properties hold trivially
GroupDiagramMiniC1.svg
2 Z2 = S2 = Dih1 simple, the smallest non-trivial group
GroupDiagramMiniC2.svg
3 Z3 = A3 simple
GroupDiagramMiniC3.svg
4 Z4 Z2  
GroupDiagramMiniC4.svg
Klein four-group, Z 2
2
 
= Dih2
Z2 (3) the smallest non-cyclic group
GroupDiagramMiniD4.svg
5 Z5 simple
GroupDiagramMiniC5.svg
6 Z6 = Z3 × Z2[1] Z3, Z2  
GroupDiagramMiniC6.svg
7 Z7 simple
GroupDiagramMiniC7.svg
8 Z8 Z4, Z2  
GroupDiagramMiniC8.svg
Z4 × Z2 Z 2
2
 
, Z4 (2), Z2 (3)
 
GroupDiagramMiniC2C4.svg
Z 3
2
 
Z 2
2
 
(7), Z2 (7)
the non-identity elements correspond to the points in the Fano plane, the Z2 × Z2 subgroups to the lines
GroupDiagramMiniC2x3.svg
9 Z9 Z3  
GroupDiagramMiniC9.svg
Z 2
3
 
Z3 (4)  
GroupDiagramMiniC3x2.svg
10 Z10 = Z5 × Z2 Z5, Z2  
GroupDiagramMiniC10.svg
11 Z11 simple
GroupDiagramMiniC11.svg
12 Z12 = Z4 × Z3 Z6, Z4, Z3, Z2  
GroupDiagramMiniC12.svg
Z6 × Z2 = Z3 × Z 2
2
 
Z6 (3), Z3, Z2 (3), Z 2
2
 
 
GroupDiagramMiniC2C6.svg
13 Z13 simple
GroupDiagramMiniC13.svg
14 Z14 = Z7 × Z2 Z7, Z2  
GroupDiagramMiniC14.svg
15 Z15 = Z5 × Z3 Z5, Z3 multiplication of nimbers 1,...,15
GroupDiagramMiniC15.svg
16 Z16 Z8, Z4, Z2  
GroupDiagramMiniC16.svg
Z 4
2
 
Z2 (15), Z 2
2
 
(35), Z 3
2
 
(15)
addition of nimbers 0,...,15
GroupDiagramMiniC2x4.svg
Z4 × Z 2
2
 
Z2 (7), Z4 (4), Z 2
2
 
(7), Z 3
2
 
, Z4 × Z2 (6)
 
GroupDiagramMiniC2x2C4.svg
Z8 × Z2 Z2 (3), Z4 (2), Z 2
2
 
, Z8 (2), Z4 × Z2
 
GroupDiagramMiniC2C8.svg
Z 2
4
 
Z2 (3), Z4 (6), Z 2
2
 
, Z4 × Z2 (3)
 
GroupDiagramMiniC4x2.svg

List of small non-abelian groups[edit]

Order Group Subgroups Properties Cycle graph
6 S3 = Dih3 Z3, Z2 (3) the smallest non-abelian group
GroupDiagramMiniD6.svg
8

Dih4

Z4, Z22 (2), Z2 (5)
GroupDiagramMiniD8.svg
quaternion group, Q8 = Dic2 Z4 (3), Z2 the smallest Hamiltonian group; smallest group demonstrating that all subgroups may be normal without the group being abelian; the smallest group G demonstrating that for a normal subgroup H the quotient group G/H need not be isomorphic to a subgroup of G
GroupDiagramMiniQ8.svg
10 Dih5 Z5, Z2 (5)
GroupDiagramMiniD10.svg
12 Dih6 = Dih3 × Z2 Z6, Dih3 (2), Z22 (3), Z3, Z2 (7)
GroupDiagramMiniD12.svg
A4 Z22, Z3 (4), Z2 (3) smallest group demonstrating that a group need not have a subgroup of every order that divides the group's order: no subgroup of order 6 (See Lagrange's theorem and the Sylow theorems.)
GroupDiagramMiniA4.svg
Dic3 = Z3 ⋊ Z4 Z2, Z3, Z4 (3), Z6
GroupDiagramMiniX12.svg
14 Dih7 Z7, Z2 (7)
GroupDiagramMiniD14.svg
16[2] Dih8 Z8, Dih4 (2), Z22 (4), Z4, Z2 (9)
GroupDiagramMiniD16.svg
Dih4 × Z2 Dih4 (2), Z4 × Z2, Z23 (2), Z22 (11), Z4 (2), Z2 (11)
GroupDiagramMiniC2D8.svg
generalized quaternion group, Q16 = Dic4
GroupDiagramMiniQ16.svg
Q8 × Z2 Hamiltonian
GroupDiagramMiniC2Q8.svg
The order 16 quasidihedral group
GroupDiagramMiniQH16.svg
The order 16 modular group
GroupDiagramMiniC2C8.svg
Z4 ⋊ Z4
GroupDiagramMinix3.svg
The group generated by the Pauli matrices
GroupDiagramMiniC2x2C4.svg
G4,4 = Z22 ⋊ Z4
GroupDiagramMiniG44.svg

Small groups library[edit]

The group theoretical computer algebra system GAP contains 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:[3]

  • 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 an additional 49487365422 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 qn × p 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.

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

See also[edit]

References[edit]

  1. ^ See a worked example showing the isomorphism Z6 = Z3 × Z2.
  2. ^ Wild, Marcel. "The Groups of Order Sixteen Made Easy", American Mathematical Monthly, Jan 2005
  3. ^ Hans Ulrich Besche The Small Groups library

External links[edit]

  • Particular groups groupprops.subwiki.org
  • Besche, H. U.; Eick, B.; O'Brien, E. "small group library". 
  • Hall, Jr., Marshall; Senior, James K. (1964). The Groups of Order 2n (n ≤ 6). Macmillan. LCCN 64016861. MR 168631. An exhaustive catalog of the 340 groups of order dividing 64 with detailed tables of defining relations, constants, and lattice presentations of each group in the notation the text defines. "Of enduring value to those interested in finite groups" (from the preface).