Under construction

Groups of small order

Abstract: The mathematical software programs GAP and MAGMA have a (common) database for small groups. This web page illustrates the cases of groups of order < 25. Groups of prime power order are omitted.

Groups of order 4

<4,1>: The cyclic group C4

<4,2>: The direct product C2 x C2

Groups of order 6

<6,1>: The symmetric group S3

<6,2>: The direct product C3 x C2

Groups of order 8

<8,1>: The cyclic group C8

<8,2>: The direct product C2 x C4

<8,3>: The dihedral group of order 8:

D8 = < a,b | a2 = b2=1, (ab)4 = 1 >.

<8,4>: The quaternion group, or dicyclic group of order 8:

Q = Q4 = < a,b | a2 = b2, b4 = 1, bab = a >.
This group has C2 x C2 as a normal subgroup.

<8,5>: The direct product C2 x C2 x C4

Groups of order 9

<9,1>: The cyclic group C9

<9,2>: The direct product C3 x C3

Groups of order 10

<10,1>: The dihedral group of order 10:

D8 = < a,b | a2 = b2=1, (ab)5 = 1 >.

<10,2>: The cyclic group C10 = C2 x C5

Groups of order 12

<12,1>: The dicyclic group of order 12:

Q6 = < a,b | a2 = b3, b6 = 1, bab = a >.

<12,2>: The cyclic group of order 12: C12

<12,3>: The alternating group of degree 4: A4

<12,4>: The dihedral group of order 12: D12

<12,5>: The direct product of two cyclic groups: C6 x C2

There are 5 isomorphism classes of groups of order 12.

Groups of order 14

<14,1>: The dihedral group of order 14:

D8 = < a,b | a2 = b2=1, (ab)7 = 1 >.

<14,2>: The cyclic group C14 = C2 x C7

Groups of order 15

<15,1>: The cyclic group C15 =C3 x C5

Groups of order 16

<16,1>: The cyclic group C16

<16,2>: The direct product of two cyclic groups: C4 x C4

<16,5>: The direct product of two cyclic groups: C8 x C2

<16,7>: The dihedral group of order 16:

D8 = < a,b | a2 = b2=1, (ab)8 = 1 >.
There are 14 isomorphism classes of groups of order 16.

Groups of order 24

<24,1>:

< a,b | a8 = 1, = b3 = 1, bab = a >.
This has C12, C6, C4, C3, C2 as normal subgroups.

<24,2> : The cyclic group of order 24: C24 = C8 x C3

<24,3>: The special linear group over GF(3),

SL(2,3) = < a,b | a4 = 1, a2 = b2, c3 = 1, aba = b , ac = cb , cab = bc >.
This has Q4, C2 as normal subgroups.

<24,4>: The dicyclic group of order 24:

Q12 = < a,b | a2 = b6, b12 = 1, bab = a >.
This group has C12, Q6, C6, C4, C3, C2 as a normal subgroup.

<24,5> : The direct product: D6 x C4 = S3 x C4

<24,6>: The dihedral group of order 24: D24.

This group has C12, D12, C6, C2 x C2, C3, C2 as a normal subgroup.

<24,7>: < a,b | a4 = 1, b6 = 1, bab = a >.

This has D12, C2 x C6, C6, C2 x C2, C3, C2 as normal subgroups.

<24,8>: < a,b,c | a3 = 1, b4 = 1, c2 = 1, bcb = c, aba = b, ac = ca >.

This has D12, Q6, C2 x C6, C2 x C2, C3, C2 as normal subgroups.

<24,9> : The direct product: C12 x C2 = C6 x C4 = C4 x C3 x C2

<24,13> : The direct product: A4 x C2

<24,14> : The direct product: D12 x C2

<24,15> : The direct product: C6 x C2x C2

last updated 11-3-2005 by wdj