# List of small groups

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

## Counts

For the number of nonisomorphic groups of order is

## Glossary

Each group is named by their Small Groups library index as G_{o}^{i}, where *o* is the order of the group, and *i* is the index of the group within that order.

Common group names:

- Z
_{n}: the cyclic group of order*n*(the notation C_{n}is also used; it is isomorphic to the additive group of**Z**/*n***Z**). - Dih
_{n}: the dihedral group of order 2*n*(often the notation D_{n}or D_{2n}is used )- K
_{4}: the Klein four-group of order 4, same as Z_{2}× Z_{2}or Dih_{2}.

- K
- 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 even permutations of*n*elements, of order 1 for*n*= 0, 1, and order*n*!/2 otherwise. - Dic
_{n}or Q_{4n}: the dicyclic group of order 4*n*.- Q
_{8}: the quaternion group of order 8, also Dic_{2}.

- Q

The notations Z_{n} and Dih_{n} have the advantage that point groups in three dimensions C_{n} and D_{n} do not have the same notation. There are more isometry groups than these two, of the same abstract group type.

The notation *G* × *H* denotes the direct product of the two groups; *G*^{n} denotes the direct product of a group with itself *n* times. *G* ⋊ *H* denotes a semidirect product where *H* acts on *G*; this may also depend on the choice of action of *H* on *G*

Abelian and simple groups are noted. (For groups of order *n* < 60, the simple groups are precisely the cyclic groups Z_{n}, 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 several isomorphic subgroups, the number of such subgroups is indicated in parentheses.

## List of small abelian groups

The finite abelian groups are either cyclic groups, or direct products thereof; see abelian groups. The numbers of nonisomorphic abelian groups of orders are

For labeled Abelian groups, see OEIS: A034382.

Order | ID | G_{o}^{i} |
Group | Nontrivial proper Subgroups | Cycle graph |
Properties |
---|---|---|---|---|---|---|

1 | 1 | G_{1}^{1} |
Z_{1} = S_{1} = A_{2} |
– | Trivial. Cyclic. Alternating. Symmetric. Elementary. | |

2 | 2 | G_{2}^{1} |
Z_{2} = S_{2} = Dih_{1} |
– | Simple. Symmetric. Cyclic. Elementary. (Smallest non-trivial group.) | |

3 | 3 | G_{3}^{1} |
Z_{3} = A_{3} |
– | Simple. Alternating. Cyclic. Elementary. | |

4 | 4 | G_{4}^{1} |
Z_{4} = Dic_{1} |
Z_{2} |
Cyclic. | |

5 | G_{4}^{2} |
Z_{2}^{2} = K_{4} = Dih_{2} |
Z_{2} (3) |
Elementary. Product. (Klein four-group. The smallest non-cyclic group.) | ||

5 | 6 | G_{5}^{1} |
Z_{5} |
– | Simple. Cyclic. Elementary. | |

6 | 8 | G_{6}^{2} |
Z_{6} = Z_{3} × Z_{2}[1] |
Z_{3}, Z_{2} |
Cyclic. Product. | |

7 | 9 | G_{7}^{1} |
Z_{7} |
– | Simple. Cyclic. Elementary. | |

8 | 10 | G_{8}^{1} |
Z_{8} |
Z_{4}, Z_{2} |
Cyclic. | |

11 | G_{8}^{2} |
Z_{4} × Z_{2} |
Z_{2}^{2}, Z_{4} (2), Z_{2} (3) |
Product. | ||

14 | G_{8}^{5} |
Z_{2}^{3} |
Z_{2}^{2} (7), Z_{2} (7) |
Product. Elementary. (The non-identity elements correspond to the points in the Fano plane, the Z_{2} × Z_{2} subgroups to the lines.) | ||

9 | 15 | G_{9}^{1} |
Z_{9} |
Z_{3} |
Cyclic. | |

16 | G_{9}^{2} |
Z_{3}^{2} |
Z_{3} (4) |
Elementary. Product. | ||

10 | 18 | G_{10}^{2} |
Z_{10} = Z_{5} × Z_{2} |
Z_{5}, Z_{2} |
Cyclic. Product. | |

11 | 19 | G_{11}^{1} |
Z_{11} |
– | Simple. Cyclic. Elementary. | |

12 | 21 | G_{12}^{2} |
Z_{12} = Z_{4} × Z_{3} |
Z_{6}, Z_{4}, Z_{3}, Z_{2} |
Cyclic. Product. | |

24 | G_{12}^{5} |
Z_{6} × Z_{2} = Z_{3} × Z_{2}^{2} |
Z_{6} (3), Z_{3}, Z_{2} (3), Z_{2}^{2} |
Product. | ||

13 | 25 | G_{13}^{1} |
Z_{13} |
– | Simple. Cyclic. Elementary. | |

14 | 27 | G_{14}^{2} |
Z_{14} = Z_{7} × Z_{2} |
Z_{7}, Z_{2} |
Cyclic. Product. | |

15 | 28 | G_{15}^{1} |
Z_{15} = Z_{5} × Z_{3} |
Z_{5}, Z_{3} |
Cyclic. Product. | |

16 | 29 | G_{16}^{1} |
Z_{16} |
Z_{8}, Z_{4}, Z_{2} |
Cyclic. | |

30 | G_{16}^{2} |
Z_{4}^{2} |
Z_{2} (3), Z_{4} (6), Z_{2}^{2}, Z_{4} × Z_{2} (3) |
Product. | ||

33 | G_{16}^{5} |
Z_{8} × Z_{2} |
Z_{2} (3), Z_{4} (2), Z_{2}^{2}, Z_{8} (2), Z_{4} × Z_{2} |
Product. | ||

38 | G_{16}^{10} |
Z_{4} × Z_{2}^{2} |
Z_{2} (7), Z_{4} (4), Z_{2}^{2} (7), Z_{2}^{3}, Z_{4} × Z_{2} (6) |
Product. | ||

42 | G_{16}^{14} |
Z_{2}^{4} = K_{4}^{2} |
Z_{2} (15), Z_{2}^{2} (35), Z_{2}^{3} (15) |
Product. Elementary. | ||

17 | 43 | G_{17}^{1} |
Z_{17} |
– | Simple. Cyclic. Elementary. | |

18 | 45 | G_{18}^{2} |
Z_{18} = Z_{9} × Z_{2} |
Z_{9}, Z_{6}, Z_{3}, Z_{2} |
Cyclic. Product. | |

48 | G_{18}^{5} |
Z_{6} × Z_{3} = Z_{3}^{2} × Z_{2} | Z_{6}, Z_{3}, Z_{2} | Product. | ||

19 | 49 | G_{19}^{1} |
Z_{19} |
– | Simple. Cyclic. Elementary. | |

20 | 51 | G_{20}^{2} |
Z_{20} = Z_{5} × Z_{4} |
Z_{10}, Z_{5}, Z_{4}, Z_{2} |
Cyclic. Product. | |

54 | G_{20}^{5} |
Z_{10} × Z_{2} = Z_{5} × Z_{2}^{2} | Z_{5}, Z_{2} |
Product. | ||

21 | 56 | G_{21}^{2} |
Z_{21} = Z_{7} × Z_{3} |
Z_{7}, Z_{3} |
Cyclic. Product. | |

22 | 58 | G_{22}^{2} |
Z_{22} = Z_{11} × Z_{2} |
Z_{11}, Z_{2} |
Cyclic. Product. | |

23 | 59 | G_{23}^{1} |
Z_{23} |
– | Simple. Cyclic. Elementary. | |

24 | 61 | G_{24}^{2} |
Z_{24} = Z_{8} × Z_{3} |
Z_{12}, Z_{8}, Z_{6}, Z_{4}, Z_{3}, Z_{2} |
Cyclic. Product. | |

68 | G_{24}^{9} |
Z_{12} × Z_{2} = Z_{6} × Z_{4}= Z _{4} × Z_{3} × Z_{2} |
Z_{12}, Z_{6}, Z_{4}, Z_{3}, Z_{2} |
Product. | ||

74 | G_{24}^{15} |
Z_{6} × Z_{2}^{2} = Z_{3} × Z_{2}^{3} |
Z_{6}, Z_{3}, Z_{2} |
Product. | ||

25 | 75 | G_{25}^{1} |
Z_{25} |
Z_{5} |
Cyclic. | |

76 | G_{25}^{2} |
Z_{5}^{2} |
Z_{5} |
Product. Elementary. | ||

26 | 78 | G_{26}^{2} |
Z_{26} = Z_{13} × Z_{2} |
Z_{13}, Z_{2} |
Cyclic. Product. | |

27 | 79 | G_{27}^{1} |
Z_{27} | Z_{9}, Z_{3} |
Cyclic. | |

80 | G_{27}^{2} |
Z_{9} × Z_{3} |
Z_{9}, Z_{3} |
Product. | ||

83 | G_{27}^{5} |
Z_{3}^{3} | Z_{3} | Product. Elementary. | ||

28 | 85 | G_{28}^{2} |
Z_{28} = Z_{7} × Z_{4} | Z_{14}, Z_{7}, Z_{4}, Z_{2} | Cyclic. Product. | |

87 | G_{28}^{4} |
Z_{14} × Z_{2} = Z_{7} × Z_{2}^{2} | Z_{14}, Z_{7}, Z_{4}, Z_{2} |
Product. | ||

29 | 88 | G_{29}^{1} |
Z_{29} |
– | Simple. Cyclic. Elementary. | |

30 | 92 | G_{30}^{4} |
Z_{30} = Z_{15} × Z_{2} = Z_{10} × Z_{3}= Z _{6} × Z_{5} = Z_{5} × Z_{3} × Z_{2} |
Z_{15}, Z_{10}, Z_{6}, Z_{5}, Z_{3}, Z_{2} |
Cyclic. Product. | |

31 | 93 | G_{31}^{1} |
Z_{31} |
– | Simple. Cyclic. Elementary. |

## 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)

Order | ID | G_{o}^{i} |
Group | Nontrivial proper Subgroups | Cycle graph |
Properties |
---|---|---|---|---|---|---|

6 | 7 | G_{6}^{1} |
Dih_{3} = S_{3} = D_{6} |
Z_{3}, Z_{2} (3) |
Dihedral group, the smallest non-abelian group, symmetric group, Frobenius group | |

8 | 12 | G_{8}^{3} |
Dih_{4} = D_{8} |
Z_{4}, Z_{2}^{2} (2), Z_{2} (5) |
Dihedral group. Extraspecial group. Nilpotent. | |

13 | G_{8}^{4} |
Q_{8} = Dic_{2} = <2,2,2> |
Z_{4} (3), Z_{2} |
Quaternion group, Hamiltonian group. all subgroups are 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. Extraspecial group Binary dihedral group. Nilpotent. | ||

10 | 17 | G_{10}^{1} |
Dih_{5} = D_{10} |
Z_{5}, Z_{2} (5) |
Dihedral group, Frobenius group | |

12 | 20 | G_{12}^{1} |
Q_{12} = Dic_{3} = <3,2,2>= Z _{3} ⋊ Z_{4} |
Z_{2}, Z_{3}, Z_{4} (3), Z_{6} |
Binary dihedral group | |

22 | G_{12}^{3} |
A_{4} |
Z_{2}^{2}, Z_{3} (4), Z_{2} (3) |
Alternating group. No subgroups of order 6, although 6 divides its order. Frobenius group | ||

23 | G_{12}^{4} |
Dih_{6} = D_{12} = Dih_{3} × Z_{2} |
Z_{6}, Dih_{3} (2), Z_{2}^{2} (3), Z_{3}, Z_{2} (7) |
Dihedral group, product | ||

14 | 26 | G_{14}^{1} |
Dih_{7} = D_{14} |
Z_{7}, Z_{2} (7) |
Dihedral group, Frobenius group | |

16[2] | 31 | G_{16}^{3} |
G_{4,4} = K_{4} ⋊ Z_{4}(Z _{4}×Z_{2}) ⋊ Z_{2} |
E_{8}, Z_{4} × Z_{2} (2), Z_{4} (4), K_{4} (6), Z_{2} (6) |
Has the same number of elements of every order as the Pauli group. Nilpotent. | |

32 | G_{16}^{4} |
Z_{4} ⋊ Z_{4} |
The squares of elements do not form a subgroup. Has the same number of elements of every order as Q_{8} × Z_{2}. Nilpotent. | |||

34 | G_{16}^{6} |
Z_{8} ⋊ Z_{2} |
Sometimes called the modular group of order 16, though this is misleading as abelian groups and Q_{8} × Z_{2} are also modular. Nilpotent. | |||

35 | G_{16}^{7} |
Dih_{8} = D_{16} |
Z_{8}, Dih_{4} (2), Z_{2}^{2} (4), Z_{4}, Z_{2} (9) |
Dihedral group. Nilpotent. | ||

36 | G_{16}^{8} |
QD_{16} |
The order 16 quasidihedral group. Nilpotent. | |||

37 | G_{16}^{9} |
Q_{16} = Dic_{4} = <4,2,2> |
generalized quaternion group, binary dihedral group. Nilpotent. | |||

39 | G_{16}^{11} |
Dih_{4} × Z_{2} |
Dih_{4} (4), Z_{4} × Z_{2}, Z_{2}^{3} (2), Z_{2}^{2} (13), Z_{4} (2), Z_{2} (11) |
Product. Nilpotent. | ||

40 | G_{16}^{12} |
Q_{8} × Z_{2} |
Hamiltonian, product. Nilpotent. | |||

41 | G_{16}^{13} |
(Z_{4} × Z_{2}) ⋊ Z_{2} |
The Pauli group generated by the Pauli matrices. Nilpotent. | |||

18 | 44 | G_{18}^{1} |
Dih_{9} = D_{18} | Dihedral group, Frobenius group | ||

46 | G_{18}^{3} |
S_{3} × Z_{3} | Product | |||

47 | G_{18}^{4} |
(Z_{3} × Z_{3}) ⋊ Z_{2} | Frobenius group | |||

20 | 50 | G_{20}^{1} |
Q_{20} = Dic_{5} = <5,2,2> | Binary dihedral group | ||

52 | G_{20}^{3} |
Z_{5} ⋊ Z_{4} | Frobenius group | |||

53 | G_{20}^{4} |
Dih_{10} = Dih_{5} × Z_{2} = D_{20} | Dihedral group, product | |||

21 | 55 | G_{21}^{1} |
Z_{7} ⋊ Z_{3} | Z_{7}, Z_{3} (7) | Smallest non-abelian group of odd order. Frobenius group | |

22 | 57 | G_{22}^{1} |
Dih_{11} = D_{22} |
Z_{11}, Z_{2} (11) |
Dihedral group, Frobenius group | |

24 | 60 | G_{24}^{1} |
Z_{3} ⋊ Z_{8} | Central extension of S_{3} | ||

62 | G_{24}^{3} |
SL(2,3) = 2T = Q_{8} ⋊ Z_{3} | Binary tetrahedral group | |||

63 | G_{24}^{4} |
Q_{24} = Dic_{6} = <6,2,2> = Z_{3} ⋊ Q_{8} | Binary dihedral | |||

64 | G_{24}^{5} |
Z_{4} × S_{3} | Product | |||

65 | G_{24}^{6} |
Dih_{12} | Dihedral group | |||

66 | G_{24}^{7} |
Dic_{3} × Z_{2} = Z_{2} × (Z_{3} ⋊ Z_{4}) | Product | |||

67 | G_{24}^{8} |
(Z_{6} × Z_{2}) ⋊ Z_{2} = Z_{3} ⋊ Dih_{4} | Double cover of dihedral group | |||

69 | G_{24}^{10} |
Dih_{4} × Z_{3} | Product. Nilpotent. | |||

70 | G_{24}^{11} |
Q_{8} × Z_{3} | Product. Nilpotent. | |||

71 | G_{24}^{12} |
S_{4} | Symmetric group. Has no normal Sylow subgroups. | |||

72 | G_{24}^{13} |
A_{4} × Z_{2} | Product | |||

73 | G_{24}^{14} |
D_{12}× Z_{2} | Product | |||

26 | 77 | G_{26}^{1} |
Dih_{13} | Dihedral group, Frobenius group | ||

27 | 81 | G_{27}^{3} |
Z_{3}^{2} ⋊ Z_{3} | All non-trivial elements have order 3. Extraspecial group. Nilpotent. | ||

82 | G_{27}^{4} |
Z_{9} ⋊ Z_{3} | Extraspecial group. Nilpotent. | |||

28 | 84 | G_{28}^{1} |
Z_{7} ⋊ Z_{4} | Binary dihedral group | ||

86 | G_{28}^{3} |
Dih_{14} | Dihedral group, product | |||

30 | 89 | G_{30}^{1} |
Z_{5} × S_{3} | Product | ||

90 | G_{30}^{2} |
Z_{3} × Dih_{5} | Product | |||

91 | G_{30}^{3} |
Dih_{15} | Dihedral group, Frobenius group |

## Classifying groups of small order

Small groups of prime power order *p*^{n} are given as follows:

- Order
*p*: The only group is cyclic. - Order
*p*^{2}: There are just 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}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
*p*^{4}: 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 S
_{4} - Order 48: The binary octahedral group and the product S
_{4}× Z_{2} - Order 60: The alternating group A
_{5}.

## Small groups library

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;
- those of cubefree order at most 50000 (395 703 groups);
- those of squarefree order;
- those of order
*p*^{n}for*n*at most 6 and*p*prime; - those of order
*p*^{7}for*p*= 3, 5, 7, 11 (907 489 groups); - those of order
*pq*^{n}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 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 SmallGroups library does not have information is 2048.

## See also

## Notes

- See a worked example showing the isomorphism Z
_{6}= Z_{3}× Z_{2}. - Wild, Marcel. "The Groups of Order Sixteen Made Easy, American Mathematical Monthly, Jan 2005
- Hans Ulrich Besche The Small Groups library Archived 2012-03-05 at the Wayback Machine

## 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 2
^{n}(n ≤ 6)". 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. Cite journal requires`|journal=`

(help)

## External links

- Particular groups in the Group Properties Wiki
- Groups of given order
- Besche, H. U.; Eick, B.; O'Brien, E. "small group library". Archived from the original on 2012-03-05.
- GroupNames database