| Group |
Order |
GAP ID (second part) |
Abelian?
|
| Trivial group |
1 |
1 |
Yes
|
| Cyclic group:Z2 |
2 |
1 |
Yes
|
| Cyclic group:Z3 |
3 |
1 |
Yes
|
| Cyclic group:Z4 |
4 |
1 |
Yes
|
| Klein four-group |
4 |
2 |
Yes
|
| Cyclic group:Z5 |
5 |
1 |
Yes
|
| Symmetric group:S3 |
6 |
2 |
No
|
| Cyclic group:Z6 |
6 |
2 |
Yes
|
| Cyclic group:Z7 |
7 |
1 |
Yes
|
| Cyclic group:Z8 |
8 |
1 |
Yes
|
| Direct product of Z4 and Z2 |
8 |
2 |
Yes
|
| Dihedral group:D8 |
8 |
3 |
No
|
| Quaternion group |
8 |
4 |
No
|
| Elementary abelian group:E8 |
8 |
5 |
Yes
|
| Cyclic group:Z9 |
9 |
1 |
Yes
|
| Elementary abelian group:E9 |
9 |
2 |
Yes
|
| Dihedral group:D10 |
10 |
1 |
No
|
| Cyclic group:Z10 |
10 |
2 |
Yes
|
| Cyclic group:Z11 |
11 |
1 |
Yes
|
| Dicyclic group:Dic12 |
12 |
1 |
No
|
| Cyclic group:Z12 |
12 |
2 |
Yes
|
| Alternating group:A4 |
12 |
3 |
No
|
| Dihedral group:D12 |
12 |
4 |
No
|
| Direct product of Z6 and Z2 |
12 |
5 |
Yes
|
| Cyclic group:Z13 |
13 |
1 |
Yes
|
| Dihedral group:D14 |
14 |
1 |
No
|
| Cyclic group:Z14 |
14 |
2 |
Yes
|
| Cyclic group:Z15 |
15 |
1 |
Yes
|
| cyclic group:Z16 |
16 |
1 |
Yes
|
| direct product of Z4 and Z4 |
16 |
2 |
Yes
|
| SmallGroup(16,3) |
16 |
3 |
No
|
| nontrivial semidirect product of Z4 and Z4 |
16 |
4 |
No
|
| direct product of Z8 and Z2 |
16 |
5 |
Yes
|
| modular maximal-cyclic group:M16 |
16 |
6 |
No
|
| dihedral group:D16 |
16 |
7 |
No
|
| semidihedral group:SD16 |
16 |
8 |
No
|
| generalized quaternion group:Q16 |
16 |
9 |
No
|
| direct product of Z4 and V4 |
16 |
10 |
Yes
|
| direct product of D8 and Z2 |
16 |
11 |
No
|
| direct product of Q8 and Z2 |
16 |
12 |
No
|
| central product of D8 and Z4 |
16 |
13 |
No
|
| elementary abelian group:E16 |
16 |
14 |
Yes
|
| Cyclic group:Z17 |
17 |
1 |
Yes
|
| dihedral group:D18 |
18 |
1 |
No
|
| cyclic group:Z18 |
18 |
2 |
Yes
|
| direct product of S3 and Z3 |
18 |
3 |
No
|
| generalized dihedral group for E9 |
18 |
4 |
No
|
| direct product of Z6 and Z3 |
18 |
5 |
Yes
|
| Cyclic group:Z19 |
19 |
1 |
Yes
|
| dicyclic group:Dic20 |
20 |
1 |
No
|
| cyclic group:Z20 |
20 |
2 |
Yes
|
| general affine group:GA(1,5) |
20 |
3 |
No
|
| dihedral group:D20 |
20 |
4 |
No
|
| direct product of Z10 and Z2 |
20 |
5 |
Yes
|
| General semilinear group:GammaL(1,8) |
21 |
1 |
No
|
| cyclic group:Z21 |
21 |
2 |
Yes
|
| dihedral group:D22 |
22 |
1 |
No
|
| cyclic group:Z22 |
22 |
2 |
Yes
|
| Cyclic group:Z23 |
23 |
1 |
Yes
|
| nontrivial semidirect product of Z3 and Z8 |
24 |
1 |
No
|
| Cyclic group:Z24 |
24 |
2 |
Yes
|
| special linear group:SL(2,3) |
24 |
3 |
No
|
| dicyclic group:Dic24 |
24 |
4 |
No
|
| direct product of S3 and Z4 |
24 |
5 |
No
|
| dihedral group:D24 |
24 |
6 |
No
|
| direct product of Dic12 and Z2 |
24 |
7 |
No
|
| SmallGroup(24,8) |
24 |
8 |
No
|
| direct product of Z6 and Z4 |
24 |
9 |
Yes
|
| direct product of D8 and Z3 |
24 |
10 |
No
|
| direct product of Q8 and Z3 |
24 |
11 |
No
|
| symmetric group:S4 |
24 |
12 |
No
|
| direct product of A4 and Z2 |
24 |
13 |
No
|
| direct product of D12 and Z2 |
24 |
14 |
No
|
| direct product of E8 and Z3 |
24 |
15 |
Yes
|
| Cyclic group:Z25 |
25 |
1 |
Yes
|
| Elementary abelian group:E25 |
25 |
2 |
Yes
|
| dihedral group:D26 |
26 |
1 |
No
|
| cyclic group:Z26 |
26 |
2 |
Yes
|
| cyclic group:Z27 |
27 |
1 |
Yes
|
| direct product of Z9 and Z3 |
27 |
2 |
Yes
|
| prime-cube order group:U(3,3) |
27 |
3 |
No
|
| M27 (semidirect product of Z9 and Z3) |
27 |
4 |
No
|
| elementary abelian group:E27 |
27 |
5 |
Yes
|
| Dicyclic group:Dic28 |
28 |
1 |
No
|
| cyclic group:Z28 |
28 |
2 |
Yes
|
| dihedral group:D28 |
28 |
3 |
No
|
| direct product of Z2 and Z14 |
28 |
4 |
Yes
|
| Cyclic group:Z29 |
29 |
1 |
Yes
|
| Direct product of Z5 and S3 |
30 |
1 |
No
|
| Direct product of Z3 and D10 |
30 |
2 |
No
|
| dihedral group:D30 |
30 |
3 |
No
|
| cyclic group:Z30 |
30 |
4 |
Yes
|
| Cyclic group:Z31 |
31 |
1 |
Yes
|
| Cyclic group:Z32 |
32 |
1 |
Yes
|
| SmallGroup(32,2) |
32 |
2 |
No
|
| Direct product of Z8 and Z4 |
32 |
3 |
Yes
|
| Semidirect product of Z8 and Z4 of M-type |
32 |
4 |
No
|
| SmallGroup(32,5) |
32 |
5 |
No
|
| Faithful semidirect product of E8 and Z4 |
32 |
6 |
No
|
| SmallGroup(32,7) |
32 |
7 |
No
|
| SmallGroup(32,8) |
32 |
8 |
No
|
| SmallGroup(32,9) |
32 |
9 |
No
|
| SmallGroup(32,10) |
32 |
10 |
No
|
| Wreath product of Z4 and Z2 |
32 |
11 |
No
|
| SmallGroup(32,12) |
32 |
12 |
No
|
| Semidirect product of Z8 and Z4 of semidihedral type |
32 |
13 |
No
|
| Semidirect product of Z8 and Z4 of dihedral type |
32 |
14 |
No
|
| SmallGroup(32,15) |
32 |
15 |
No
|
| Direct product of Z16 and Z2 |
32 |
16 |
Yes
|
| M32 |
32 |
17 |
No
|
| Dihedral group:D32 |
32 |
18 |
No
|
| Semidihedral group:SD32 |
32 |
19 |
No
|
| Generalized quaternion group:Q32 |
32 |
20 |
No
|
| Direct product of Z4 and Z4 and Z2 |
32 |
21 |
Yes
|
| Direct product of SmallGroup(16,3) and Z2 |
32 |
22 |
No
|
| Direct product of SmallGroup(16,4) and Z2 |
32 |
23 |
No
|
| SmallGroup(32,24) |
32 |
24 |
No
|
| Direct product of D8 and Z4 |
32 |
25 |
No
|
| Direct product of Q8 and Z4 |
32 |
26 |
No
|
| SmallGroup(32,27) |
32 |
27 |
No
|
| SmallGroup(32,28) |
32 |
28 |
No
|
| SmallGroup(32,29) |
32 |
29 |
No
|
| SmallGroup(32,30) |
32 |
30 |
No
|
| SmallGroup(32,31) |
32 |
31 |
No
|
| SmallGroup(32,32) |
32 |
32 |
No
|
| SmallGroup(32,33) |
32 |
33 |
No
|
| Generalized dihedral group for direct product of Z4 and Z4 |
32 |
34 |
No
|
| SmallGroup(32,35) |
32 |
35 |
No
|
| Direct product of Z8 and V4 |
32 |
36 |
Yes
|
| Direct product of M16 and Z2 |
32 |
37 |
No
|
| Central product of D8 and Z8 |
32 |
38 |
No
|
| Direct product of D16 and Z2 |
32 |
39 |
No
|
| Direct product of SD16 and Z2 |
32 |
40 |
No
|
| Direct product of Q16 and Z2 |
32 |
41 |
No
|
| Central product of D16 and Z4 |
32 |
42 |
No
|
| Holomorph of Z8 |
32 |
43 |
No
|
| SmallGroup(32,44) |
32 |
44 |
No
|
| Direct product of E8 and Z4 |
32 |
45 |
Yes
|
| Direct product of D8 and V4 |
32 |
46 |
No
|
| Direct product of Q8 and V4 |
32 |
47 |
No
|
| Direct product of SmallGroup(16,13) and Z2 |
32 |
48 |
No
|
| Inner holomorph of D8 |
32 |
49 |
No
|
| Central product of D8 and Q8 |
32 |
50 |
No
|
| Elementary abelian group:E32 |
32 |
51 |
Yes
|
| Cyclic group:Z33 |
33 |
1 |
Yes
|
| dihedral group:D34 |
34 |
1 |
No
|
| cyclic group:Z34 |
34 |
2 |
Yes
|
| Cyclic group:Z35 |
35 |
1 |
Yes
|
| Dicyclic group:Dic36 |
36 |
1 |
No
|
| Cyclic group:Z36 |
36 |
2 |
Yes
|
| SmallGroup(36,3) |
36 |
3 |
No
|
| Dihedral group:D36 |
36 |
4 |
No
|
| Direct product of E4 and Z9 |
36 |
5 |
Yes
|
| SmallGroup(36,6) |
36 |
6 |
No
|
| SmallGroup(36,7) |
36 |
7 |
No
|
| Direct product of E9 and Z4 |
36 |
8 |
Yes
|
| SmallGroup(36,9) |
36 |
9 |
No
|
| Direct product of S3 and S3 |
36 |
10 |
No
|
| Direct product of A4 and Z3 |
36 |
11 |
No
|
| Direct product of D12 and Z3 |
36 |
12 |
No
|
| SmallGroup(36,13) |
36 |
13 |
No
|
| Direct product of E4 and E9 |
36 |
14 |
Yes
|
| Cyclic group:Z37 |
37 |
1 |
Yes
|
| dihedral group:D38 |
38 |
1 |
No
|
| cyclic group:Z38 |
38 |
2 |
Yes
|
| Semidirect product of Z13 with Z3 |
39 |
1 |
No
|
| cyclic group:Z39 |
39 |
2 |
Yes
|
| semidirect product of Z5 and Z8 via inverse map |
40 |
1 |
No
|
| cyclic group:Z40 |
40 |
2 |
Yes
|
| semidirect product of Z5 and Z8 via square map |
40 |
3 |
No
|
| nontrivial semidirect product of Z5 and Q8 |
40 |
4 |
No
|
| direct product of D10 and Z4 |
40 |
5 |
No
|
| dihedral group:D40 |
40 |
6 |
No
|
| SmallGroup(40,7) |
40 |
7 |
No
|
| SmallGroup(40,8) |
40 |
8 |
No
|
| direct product of Z20 and Z2 (also direct product of Z10 and Z4) |
40 |
9 |
Yes
|
| direct product of D8 and Z5 |
40 |
10 |
No
|
| direct product of Q8 and Z5 |
40 |
11 |
No
|
| direct product of GA(1,5) and Z2 |
40 |
12 |
No
|
| direct product of D10 and V4 |
40 |
13 |
No
|
| direct product of E8 and Z5 |
40 |
14 |
Yes
|
| Cyclic group:Z41 |
41 |
1 |
Yes
|
| General affine group:GA(1,7), also known as F7 |
42 |
1 |
No
|
| direct product of Z2 and the semidirect product of Z7 and Z3 |
42 |
2 |
No
|
| direct product of S3 and Z7 |
42 |
3 |
No
|
| direct product of D14 and Z3 |
42 |
4 |
No
|
| dihedral group:D42 |
42 |
5 |
No
|
| cyclic group:Z42 |
42 |
6 |
Yes
|
| Cyclic group:Z43 |
43 |
1 |
Yes
|
| Dicyclic group:Dic44 |
44 |
1 |
No
|
| cyclic group:Z44 |
44 |
2 |
Yes
|
| dihedral group:D44 |
44 |
3 |
No
|
| direct product of Z2 and Z22 |
44 |
4 |
Yes
|
| cyclic group:Z45 |
45 |
1 |
Yes
|
| direct product of Z15 and Z3 |
45 |
2 |
Yes
|
| dihedral group:D46 |
46 |
1 |
No
|
| cyclic group:Z46 |
46 |
2 |
Yes
|
| Cyclic group:Z47 |
47 |
1 |
Yes
|
| SmallGroup(48,1) |
48 |
1 |
No
|
| Cyclic group:Z48 |
48 |
2 |
Yes
|
| SmallGroup(48,3) |
48 |
3 |
No
|
| SmallGroup(48,4) |
48 |
4 |
No
|
| SmallGroup(48,5) |
48 |
5 |
No
|
| SmallGroup(48,6) |
48 |
6 |
No
|
| Dihedral group:D48 |
48 |
7 |
No
|
| SmallGroup(48,8) |
48 |
8 |
No
|
| SmallGroup(48,9) |
48 |
9 |
No
|
| SmallGroup(48,10) |
48 |
10 |
No
|
| SmallGroup(48,11) |
48 |
11 |
No
|
| SmallGroup(48,12) |
48 |
12 |
No
|
| SmallGroup(48,13) |
48 |
13 |
No
|
| SmallGroup(48,14) |
48 |
14 |
No
|
| SmallGroup(48,15) |
48 |
15 |
No
|
| SmallGroup(48,16) |
48 |
16 |
No
|
| SmallGroup(48,17) |
48 |
17 |
No
|
| SmallGroup(48,18) |
48 |
18 |
No
|
| SmallGroup(48,19) |
48 |
19 |
No
|
| SmallGroup(48,20) |
48 |
20 |
Yes
|
| Direct product of SmallGroup(16,3) and Z3 |
48 |
21 |
No
|
| SmallGroup(48,22) |
48 |
22 |
No
|
| SmallGroup(48,23) |
48 |
23 |
Yes
|
| Direct product of M16 and Z3 |
48 |
24 |
No
|
| Direct product of D16 and Z3 |
48 |
25 |
No
|
| Direct product of SD16 and Z3 |
48 |
26 |
No
|
| SmallGroup(48,27) |
48 |
27 |
No
|
| Binary octahedral group |
48 |
28 |
No
|
| General linear group:GL(2,3) |
48 |
29 |
No
|
| Special linear group:SL(2,Z4) |
48 |
30 |
No
|
| Direct product of A4 and Z4 |
48 |
31 |
No
|
| Direct product of SL(2,3) and Z2 |
48 |
32 |
No
|
| Central product of SL(2,3) and Z4 |
48 |
33 |
No
|
| SmallGroup(48,34) |
48 |
34 |
No
|
| SmallGroup(48,35) |
48 |
35 |
No
|
| SmallGroup(48,36) |
48 |
36 |
No
|
| SmallGroup(48,37) |
48 |
37 |
No
|
| Direct product of D8 and S3 |
48 |
38 |
No
|
| SmallGroup(48,39) |
48 |
39 |
No
|
| Direct product of Q8 and S3 |
48 |
40 |
No
|
| SmallGroup(48,41) |
48 |
41 |
No
|
| SmallGroup(48,42) |
48 |
42 |
No
|
| SmallGroup(48,43) |
48 |
43 |
No
|
| SmallGroup(48,44) |
48 |
44 |
Yes
|
| Direct product of D8 and Z6 |
48 |
45 |
No
|
| SmallGroup(48,46) |
48 |
46 |
No
|
| Central product of D8 and Z12 |
48 |
47 |
No
|
| Direct product of S4 and Z2 |
48 |
48 |
No
|
| Direct product of A4 and V4 |
48 |
49 |
No
|
| SmallGroup(48,50) |
48 |
50 |
No
|
| SmallGroup(48,51) |
48 |
51 |
No
|
| SmallGroup(48,52) |
48 |
52 |
Yes
|
| Cyclic group:Z49 |
49 |
1 |
Yes
|
| Elementary abelian group:E49 |
49 |
2 |
Yes
|
| dihedral group:D50 |
50 |
1 |
No
|
| cyclic group:Z50 |
50 |
2 |
Yes
|
| direct product of C5 and D10 |
50 |
3 |
No
|
| semidirect product of direct product of Z5 and Z5 with Z2 |
50 |
4 |
No
|
| direct product of Z10 and C5 |
50 |
5 |
Yes
|