The list
| Group |
Second part of GAP ID (GAP ID is (32,second part)) |
Hall-Senior number (among groups of order 16) |
Hall-Senior symbol
|
| Cyclic group:Z32 |
1 |
7 |
|
| SmallGroup(32,2) |
2 |
|
|
| Direct product of Z8 and Z4 |
3 |
|
|
| SmallGroup(32,4) |
4 |
|
|
| SmallGroup(32,5) |
5 |
|
|
| Faithful semidirect product of E8 and Z4 |
6 |
|
|
| SmallGroup(32,7) |
7 |
|
|
| SmallGroup(32,8) |
8 |
|
|
| SmallGroup(32,9) |
9 |
|
|
| SmallGroup(32,10) |
10 |
|
|
| Wreath product of Z4 and Z2 |
11 |
|
|
| SmallGroup(32,12) |
12 |
|
|
| SmallGroup(32,13) |
13 |
|
|
| SmallGroup(32,14) |
14 |
|
|
| SmallGroup(32,15) |
15 |
|
|
| Direct product of Z16 and Z2 |
16 |
|
|
| M32 |
17 |
|
|
| Dihedral group:D32 |
18 |
|
|
| Semidihedral group:SD32 |
19 |
|
|
| Generalized quaternion group:Q32 |
20 |
|
|
| SmallGroup(32,21) |
21 |
|
|
| Direct product of SmallGroup(16,3) and Z2 |
22 |
|
|
| Direct product of SmallGroup(16,4) and Z2 |
23 |
|
|
| SmallGroup(32,24) |
24 |
|
|
| Direct product of D8 and Z4 |
25 |
|
|
| Direct product of Q8 and Z4 |
26 |
|
|
| SmallGroup(32,27) |
27 |
|
|
| SmallGroup(32,28) |
28 |
|
|
| SmallGroup(32,29) |
29 |
|
|
| SmallGroup(32,30) |
30 |
|
|
| SmallGroup(32,31) |
31 |
|
|
| SmallGroup(32,32) |
32 |
|
|
| SmallGroup(32,33) |
33 |
|
|
| Generalized dihedral group for direct product of Z4 and Z4 |
34 |
|
|
| SmallGroup(32,35) |
35 |
|
|
| Direct product of Z8 and V4 |
36 |
|
|
| Direct product of M16 and Z2 |
37 |
|
|
| SmallGroup(32,38) |
38 |
|
|
| Direct product of D16 and Z2 |
39 |
|
|
| Direct product of SD16 and Z2 |
40 |
|
|
| SmallGroup(32,41) |
41 |
|
|
| SmallGroup(32,42) |
42 |
|
|
| Holomorph of Z8 |
43 |
|
|
| SmallGroup(32,44) |
44 |
|
|
| Direct product of E8 and Z4 |
45 |
|
|
| Direct product of D8 and V4 |
46 |
|
|
| Direct product of Q8 and V4 |
47 |
|
|
| SmallGroup(32,48) |
48 |
|
|
| Inner holomorph of D8 |
49 |
|
|
| SmallGroup(32,50) |
50 |
|
|
| Elementary abelian group:E32 |
51 |
1 |
|
Element structure
Order statistics
FACTS TO CHECK AGAINST:
ORDER STATISTICS (cf. order statistics, order statistics-equivalent finite groups): number of nth roots is a multiple of n | Finite abelian groups with the same order statistics are isomorphic | Lazard Lie group has the same order statistics as the additive group of its Lazard Lie ring | Frobenius conjecture on nth roots
1-ISOMORPHISM (cf. 1-isomorphic groups): Lazard Lie group is 1-isomorphic to the additive group of its Lazard Lie ring | order statistics-equivalent not implies 1-isomorphic
Note that because number of nth roots is a multiple of n, we see that the number of elements whose order is or is odd, while all the other numbers are even. The total number of roots is even for all .
| Group |
Second part of GAP ID |
Hall-Senior number |
Number of elements of order 1 |
Number of elements of order 2 |
Number of elements of order 4 |
Number of elements of order 8 |
Number of elements of order 16 |
Number of elements of order 32
|
| Cyclic group:Z32 |
1 |
|
1 |
1 |
2 |
4 |
8 |
16
|
| SmallGroup(32,2) |
2 |
|
1 |
7 |
24 |
0 |
0 |
0
|
| Direct product of Z8 and Z4 |
3 |
|
1 |
3 |
12 |
16 |
0 |
0
|
| SmallGroup(32,4) |
4 |
|
1 |
3 |
12 |
16 |
0 |
0
|
| SmallGroup(32,5) |
5 |
|
1 |
7 |
8 |
16 |
0 |
0
|
| Faithful semidirect product of E8 and Z4 |
6 |
|
1 |
11 |
20 |
0 |
0 |
0
|
| SmallGroup(32,7) |
7 |
|
1 |
11 |
4 |
16 |
0 |
0
|
| SmallGroup(32,8) |
8 |
|
1 |
3 |
12 |
16 |
0 |
0
|
| SmallGroup(32,9) |
9 |
|
1 |
11 |
12 |
8 |
0 |
0
|
| SmallGroup(32,10) |
10 |
|
1 |
3 |
20 |
8 |
0 |
0
|
| Wreath product of Z4 and Z2 |
11 |
|
1 |
7 |
16 |
8 |
0 |
0
|
| SmallGroup(32,12) |
12 |
|
1 |
3 |
12 |
16 |
0 |
0
|
| SmallGroup(32,13) |
13 |
|
1 |
3 |
20 |
8 |
0 |
0
|
| SmallGroup(32,14) |
14 |
|
1 |
3 |
20 |
8 |
0 |
0
|
| SmallGroup(32,15) |
15 |
|
1 |
3 |
4 |
24 |
0 |
0
|
| Direct product of Z16 and Z2 |
16 |
|
1 |
3 |
4 |
8 |
16 |
0
|
| M32 |
17 |
|
1 |
3 |
4 |
8 |
16 |
0
|
| Dihedral group:D32 |
18 |
|
1 |
17 |
2 |
4 |
8 |
0
|
| Semidihedral group:SD32 |
19 |
|
1 |
9 |
10 |
4 |
8 |
0
|
| Generalized quaternion group:Q32 |
20 |
|
1 |
1 |
18 |
4 |
8 |
0
|
| SmallGroup(32,21) |
21 |
|
1 |
7 |
24 |
0 |
0 |
0
|
| Direct product of SmallGroup(16,3) and Z2 |
22 |
|
1 |
15 |
16 |
0 |
0
|
| Direct product of SmallGroup(16,4) and Z2 |
23 |
|
1 |
7 |
24 |
0 |
0
|
| SmallGroup(32,24) |
24 |
|
1 |
7 |
24 |
0 |
0 |
0
|
| Direct product of D8 and Z4 |
25 |
|
1 |
11 |
20 |
0 |
0 |
0
|
| Direct product of Q8 and Z4 |
26 |
|
1 |
3 |
28 |
0 |
0 |
0
|
| SmallGroup(32,27) |
27 |
|
1 |
19 |
12 |
0 |
0 |
0
|
| SmallGroup(32,28) |
28 |
|
1 |
15 |
16 |
0 |
0 |
0
|
| SmallGroup(32,29) |
29 |
|
1 |
7 |
24 |
0 |
0 |
0
|
| SmallGroup(32,30) |
30 |
|
1 |
11 |
20 |
0 |
0 |
0
|
| SmallGroup(32,31) |
31 |
|
1 |
11 |
20 |
0 |
0 |
0
|
| SmallGroup(32,32) |
32 |
|
1 |
3 |
28 |
0 |
0 |
0
|
| SmallGroup(32,33) |
33 |
|
1 |
7 |
24 |
0 |
0 |
0
|
| Generalized dihedral group for direct product of Z4 and Z4 |
34 |
|
1 |
19 |
12 |
0 |
0 |
0
|
| SmallGroup(32,35) |
35 |
|
1 |
3 |
28 |
0 |
0 |
0
|
| Direct product of Z8 and V4 |
36 |
|
1 |
7 |
8 |
16 |
0 |
0
|
| Direct product of M16 and Z2 |
37 |
|
1 |
7 |
8 |
16 |
0 |
0
|
| SmallGroup(32,38) |
38 |
|
1 |
7 |
8 |
16 |
0 |
0
|
| Direct product of D16 and Z2 |
39 |
|
1 |
19 |
4 |
8 |
0 |
0
|
| Direct product of SD16 and Z2 |
40 |
|
1 |
11 |
12 |
8 |
0 |
0
|
| SmallGroup(32,41) |
41 |
|
1 |
3 |
20 |
8 |
0 |
0
|
| SmallGroup(32,42) |
42 |
|
1 |
11 |
12 |
8 |
0 |
0
|
| Holomorph of Z8 |
43 |
|
1 |
15 |
8 |
8 |
0 |
0
|
| SmallGroup(32,44) |
44 |
|
1 |
7 |
16 |
8 |
0 |
0
|
| Direct product of E8 and Z4 |
45 |
|
1 |
15 |
16 |
0 |
0 |
0
|
| Direct product of D8 and V4 |
46 |
|
1 |
23 |
8 |
0 |
0 |
0
|
| Direct product of Q8 and V4 |
47 |
|
1 |
7 |
24 |
0 |
0 |
0
|
| SmallGroup(32,48) |
48 |
|
1 |
1 |
15 |
16 |
0 |
0 |
0
|
| Inner holomorph of D8 |
49 |
|
1 |
19 |
12 |
0 |
0 |
0
|
| SmallGroup(32,50) |
50 |
|
1 |
11 |
20 |
0 |
0 |
0
|
| Elementary abelian group:E32 |
51 |
|
1 |
31 |
0 |
0 |
0 |
0
|