small groups by order

Order Id Structure decription Group Name Characteristic
1 [1, 1] 1  Trivial group
2 [2, 1] C2  Cyclic group 2
3 [3, 1] C3  Cyclic group 3
4 [4, 1] C4  Cyclic group 2
4 [4, 2] C2\(\times\)C2  Klein 4-group V4 = elementary abelian group of type [2, 2] 2
5 [5, 1] C5  Cyclic group 5
6 [6, 1] S3  Symmetric group on 3 letters 2, 3
6 [6, 2] C6  Cyclic group 2, 3
7 [7, 1] C7  Cyclic group 7
8 [8, 1] C8  Cyclic group 2
8 [8, 2] C4\(\times\)C2  Abelian group of type [2, 4] 2
8 [8, 3] D8  Dihedral group 2
8 [8, 4] Q8  Quaternion group 2
8 [8, 5] C2\(\times\)C2\(\times\)C2  Elementary abelian group of type [2, 2,2] 2
9 [9, 1] C9  Cyclic group 3
9 [9, 2] C3\(\times\)C3  Elementary abelian group of type [3, 3] 3
10 [10, 1] D10  Dihedral group 2, 5
10 [10, 2] C10  Cyclic group 2, 5
11 [11, 1] C11  Cyclic group 11
12 [12, 1] C3 \(\rtimes\) C4  Dicyclic group 2, 3
12 [12, 2] C12  Cyclic group 2, 3
12 [12, 3] A4  Alternating group on 4 letters 2, 3
12 [12, 4] D12  Dihedral group 2, 3
12 [12, 5] C6\(\times\)C2  Abelian group of type [2, 6] 2, 3
13 [13, 1] C13  Cyclic group 13
14 [14, 1] D14  Dihedral group 2, 7
14 [14, 2] C14  Cyclic group 2, 7
15 [15, 1] C15  Cyclic group 3, 5
16 [16, 1] C16  Cyclic group 2
16 [16, 2] C4\(\times\)C4  Abelian group of type [4, 4] 2
16 [16, 3] (C4 \(\times\)C2)\(\rtimes\) C2  The semidirect product of C22 and C4 acting via C4/C2=C2 2
16 [16, 4] C4 \(\rtimes\) C4  The semidirect product of C4 and C4 acting via C4/C2=C2 2
16 [16, 5] C8\(\times\)C2  Abelian group of type [2, 8] 2
16 [16, 6] C8 \(\rtimes\) C2  Modular maximal-cyclic group 2
16 [16, 7] D16  Dihedral group 2
16 [16, 8] QD16  Semidihedral group 2
16 [16, 9] Q16  Generalised quaternion group 2
16 [16, 10] C4\(\times\)C2\(\times\)C2  Abelian group of type [2, 2,4] 2
16 [16, 11] C2\(\times\)D8  Direct product of C2 and D4 2
16 [16, 12] C2\(\times\)Q8  Direct product of C2 and Q8 2
16 [16, 13] (C4 \(\times\)C2)\(\rtimes\) C2  Pauli group = central product of C4 and D4 2
16 [16, 14] C2\(\times\)C2\(\times\)C2\(\times\)C2  Elementary abelian group of type [2, 2,2, 2] 2
17 [17, 1] C17  Cyclic group 17
18 [18, 1] D18  Dihedral group 2, 3
18 [18, 2] C18  Cyclic group 2, 3
18 [18, 3] C3\(\times\)S3  Direct product of C3 and S3 2, 3
18 [18, 4] (C3 \(\times\)C3)\(\rtimes\) C2  The semidirect product of C3 and S3 acting via S3/C3=C2 2, 3
18 [18, 5] C6\(\times\)C3  Abelian group of type [3, 6] 2, 3
19 [19, 1] C19  Cyclic group 19
20 [20, 1] C5 \(\rtimes\) C4  Dicyclic group 2, 5
20 [20, 2] C20  Cyclic group 2, 5
20 [20, 3] C5 \(\rtimes\) C4  Frobenius group 2, 5
20 [20, 4] D20  Dihedral group 2, 5
20 [20, 5] C10\(\times\)C2  Abelian group of type [2, 10] 2, 5
21 [21, 1] C7 \(\rtimes\) C3  The semidirect product of C7 and C3 acting faithfully 3, 7
21 [21, 2] C21  Cyclic group 3, 7
22 [22, 1] D22  Dihedral group 2, 11
22 [22, 2] C22  Cyclic group 2, 11
23 [23, 1] C23  Cyclic group 23
24 [24, 1] C3 \(\rtimes\) C8  The semidirect product of C3 and C8 acting via C8/C4=C2 2, 3
24 [24, 2] C24  Cyclic group 2, 3
24 [24, 3] SL(2, 3)  Special linear group on 𝔽32 2, 3
24 [24, 4] C3 \(\rtimes\) Q8  Dicyclic group 2, 3
24 [24, 5] C4\(\times\)S3  Direct product of C4 and S3 2, 3
24 [24, 6] D24  Dihedral group 2, 3
24 [24, 7] C2\(\times\)(C3  \(\rtimes\) C4) Direct product of C2 and Dic3 2, 3
24 [24, 8] (C6 \(\times\)C2)\(\rtimes\) C2  The semidirect product of C3 and D4 acting via D4/C22=C2 2, 3
24 [24, 9] C12\(\times\)C2  Abelian group of type [2, 12] 2, 3
24 [24, 10] C3\(\times\)D8  Direct product of C3 and D4 2, 3
24 [24, 11] C3\(\times\)Q8  Direct product of C3 and Q8 2, 3
24 [24, 12] S4  Symmetric group on 4 letters 2, 3
24 [24, 13] C2\(\times\)A4  Direct product of C2 and A4 2, 3
24 [24, 14] C2\(\times\)C2\(\times\)S3  Direct product of C22 and S3 2, 3
24 [24, 15] C6\(\times\)C2\(\times\)C2  Abelian group of type [2, 2,6] 2, 3
25 [25, 1] C25  Cyclic group 5
25 [25, 2] C5\(\times\)C5  Elementary abelian group of type [5, 5] 5
26 [26, 1] D26  Dihedral group 2, 13
26 [26, 2] C26  Cyclic group 2, 13
27 [27, 1] C27  Cyclic group 3
27 [27, 2] C9\(\times\)C3  Abelian group of type [3, 9] 3
27 [27, 3] (C3 \(\times\)C3)\(\rtimes\) C3  Heisenberg group 3
27 [27, 4] C9 \(\rtimes\) C3  Extraspecial group 3
27 [27, 5] C3\(\times\)C3\(\times\)C3  Elementary abelian group of type [3, 3,3] 3
28 [28, 1] C7 \(\rtimes\) C4  Dicyclic group 2, 7
28 [28, 2] C28  Cyclic group 2, 7
28 [28, 3] D28  Dihedral group 2, 7
28 [28, 4] C14\(\times\)C2  Abelian group of type [2, 14] 2, 7
29 [29, 1] C29  Cyclic group 29
30 [30, 1] C5\(\times\)S3  Direct product of C5 and S3 2, 3, 5
30 [30, 2] C3\(\times\)D10  Direct product of C3 and D5 2, 3, 5
30 [30, 3] D30  Dihedral group 2, 3, 5
30 [30, 4] C30  Cyclic group 2, 3, 5
31 [31, 1] C31  Cyclic group 31
32 [32, 1] C32  Cyclic group 2
32 [32, 2] (C4 \(\times\)C2)\(\rtimes\) C4  1st central stem extension by C2 of C42 2
32 [32, 3] C8\(\times\)C4  Abelian group of type [4, 8] 2
32 [32, 4] C8 \(\rtimes\) C4  3rd semidirect product of C8 and C4 acting via C4/C2=C2 2
32 [32, 5] (C8 \(\times\)C2)\(\rtimes\) C2  The semidirect product of C22 and C8 acting via C8/C4=C2 2
32 [32, 6] (C2 \(\times\)C2\(\times\)C2)\(\rtimes\) C4  The semidirect product of C23 and C4 acting faithfully 2
32 [32, 7] (C8  \(\rtimes\) C2)\(\rtimes\) C2  1st non-split extension by C4 of D4 acting via D4/C22=C2 2
32 [32, 8] C2.((C4\(\times\)C2)\(\rtimes\) C2)= (C2 \(\times\)C2). (C4 \(\times\)C2) 2nd non-split extension by C4 of D4 acting via D4/C22=C2 2
32 [32, 9] (C8 \(\times\)C2)\(\rtimes\) C2  1st semidirect product of D4 and C4 acting via C4/C2=C2 2
32 [32, 10] Q8 \(\rtimes\) C4  1st semidirect product of Q8 and C4 acting via C4/C2=C2 2
32 [32, 11] (C4 \(\times\)C4)\(\rtimes\) C2  Wreath product of C4 by C2 2
32 [32, 12] C4 \(\rtimes\) C8  The semidirect product of C4 and C8 acting via C8/C4=C2 2
32 [32, 13] C8 \(\rtimes\) C4  1st non-split extension by C4 of Q8 acting via Q8/C4=C2 2
32 [32, 14] C8 \(\rtimes\) C4  2nd central extension by C2 of D8 2
32 [32, 15] C4.D8 = C4.(C4 \(\times\)C2) 1st non-split extension by C8 of C4 acting via C4/C2=C2 2
32 [32, 16] C16\(\times\)C2  Abelian group of type [2, 16] 2
32 [32, 17] C16 \(\rtimes\) C2  Modular maximal-cyclic group 2
32 [32, 18] D32  Dihedral group 2
32 [32, 19] QD32  Semidihedral group 2
32 [32, 20] Q32  Generalised quaternion group 2
32 [32, 21] C4\(\times\)C4\(\times\)C2  Abelian group of type [2, 4,4] 2
32 [32, 22] C2\(\times\)((C4\(\times\)C2)\(\rtimes\) C2) Direct product of C2 and C22\(\rtimes\)C4 2
32 [32, 23] C2\(\times\)(C4  \(\rtimes\) C4) Direct product of C2 and C4\(\rtimes\)C4 2
32 [32, 24] (C4 \(\times\)C4)\(\rtimes\) C2  1st semidirect product of C42 and C2 acting faithfully 2
32 [32, 25] C4\(\times\)D8  Direct product of C4 and D4 2
32 [32, 26] C4\(\times\)Q8  Direct product of C4 and Q8 2
32 [32, 27] (C2 \(\times\)C2\(\times\)C2\(\times\)C2)\(\rtimes\) C2  Wreath product of C22 by C2 2
32 [32, 28] (C4 \(\times\)C2\(\times\)C2)\(\rtimes\) C2  The semidirect product of C4 and D4 acting via D4/C22=C2 2
32 [32, 29] (C2 \(\times\)Q8)\(\rtimes\) C2  The semidirect product of C22 and Q8 acting via Q8/C4=C2 2
32 [32, 30] (C4 \(\times\)C2\(\times\)C2)\(\rtimes\) C2  3rd non-split extension by C22 of D4 acting via D4/C22=C2 2
32 [32, 31] (C4 \(\times\)C4)\(\rtimes\) C2  4th non-split extension by C4 of D4 acting via D4/C4=C2 2
32 [32, 32] (C2 \(\times\)C2). (C2 \(\times\)C2\(\times\)C2) 4th non-split extension by C42 of C2 acting faithfully 2
32 [32, 33] (C4 \(\times\)C4)\(\rtimes\) C2  2nd semidirect product of C42 and C2 acting faithfully 2
32 [32, 34] (C4 \(\times\)C4)\(\rtimes\) C2  The semidirect product of C4 and D4 acting via D4/C4=C2 2
32 [32, 35] C4 \(\rtimes\) Q8  The semidirect product of C4 and Q8 acting via Q8/C4=C2 2
32 [32, 36] C8\(\times\)C2\(\times\)C2  Abelian group of type [2, 2,8] 2
32 [32, 37] C2\(\times\)(C8  \(\rtimes\) C2) Direct product of C2 and M4(2) 2
32 [32, 38] (C8 \(\times\)C2)\(\rtimes\) C2  Central product of C8 and D4 2
32 [32, 39] C2\(\times\)D16  Direct product of C2 and D8 2
32 [32, 40] C2\(\times\)QD16  Direct product of C2 and SD16 2
32 [32, 41] C2\(\times\)Q16  Direct product of C2 and Q16 2
32 [32, 42] (C8 \(\times\)C2)\(\rtimes\) C2  Central product of C4 and D8 2
32 [32, 43] C8 \(\rtimes\) (C2 \(\times\)C2) The semidirect product of C8 and C22 acting faithfully 2
32 [32, 44] (C2 \(\times\)Q8)\(\rtimes\) C2  The non-split extension by C8 of C22 acting faithfully 2
32 [32, 45] C4\(\times\)C2\(\times\)C2\(\times\)C2  Abelian group of type [2, 2,2, 4] 2
32 [32, 46] C2\(\times\)C2\(\times\)D8  Direct product of C22 and D4 2
32 [32, 47] C2\(\times\)C2\(\times\)Q8  Direct product of C22 and Q8 2
32 [32, 48] C2\(\times\)((C4\(\times\)C2)\(\rtimes\) C2) Direct product of C2 and C4○D4 2
32 [32, 49] (C2 \(\times\)C2\(\times\)C2)\(\rtimes\) (C2 \(\times\)C2) Extraspecial group 2
32 [32, 50] (C2 \(\times\)Q8)\(\rtimes\) C2  Gamma matrices = Extraspecial group 2
32 [32, 51] C2\(\times\)C2\(\times\)C2\(\times\)C2\(\times\)C2  Elementary abelian group of type [2, 2,2, 2,2] 2
33 [33, 1] C33  Cyclic group 3, 11
34 [34, 1] D34  Dihedral group 2, 17
34 [34, 2] C34  Cyclic group 2, 17
35 [35, 1] C35  Cyclic group 5, 7
36 [36, 1] C9 \(\rtimes\) C4  Dicyclic group 2, 3
36 [36, 2] C36  Cyclic group 2, 3
36 [36, 3] (C2 \(\times\)C2)\(\rtimes\) C9  The central extension by C3 of A4 2, 3
36 [36, 4] D36  Dihedral group 2, 3
36 [36, 5] C18\(\times\)C2  Abelian group of type [2, 18] 2, 3
36 [36, 6] C3\(\times\)(C3  \(\rtimes\) C4) Direct product of C3 and Dic3 2, 3
36 [36, 7] (C3 \(\times\)C3)\(\rtimes\) C4  The semidirect product of C3 and Dic3 acting via Dic3/C6=C2 2, 3
36 [36, 8] C12\(\times\)C3  Abelian group of type [3, 12] 2, 3
36 [36, 9] (C3 \(\times\)C3)\(\rtimes\) C4  The semidirect product of C32 and C4 acting faithfully 2, 3
36 [36, 10] S3\(\times\)S3  Direct product of S3 and S3 2, 3
36 [36, 11] C3\(\times\)A4  Direct product of C3 and A4 2, 3
36 [36, 12] C6\(\times\)S3  Direct product of C6 and S3 2, 3
36 [36, 13] C2\(\times\)((C3\(\times\)C3)\(\rtimes\) C2) Direct product of C2 and C3\(\rtimes\)S3 2, 3
36 [36, 14] C6\(\times\)C6  Abelian group of type [6, 6] 2, 3
37 [37, 1] C37  Cyclic group 37
38 [38, 1] D38  Dihedral group 2, 19
38 [38, 2] C38  Cyclic group 2, 19
39 [39, 1] C13 \(\rtimes\) C3  The semidirect product of C13 and C3 acting faithfully 3, 13
39 [39, 2] C39  Cyclic group 3, 13
40 [40, 1] C5 \(\rtimes\) C8  The semidirect product of C5 and C8 acting via C8/C4=C2 2, 5
40 [40, 2] C40  Cyclic group 2, 5
40 [40, 3] C5 \(\rtimes\) C8  The semidirect product of C5 and C8 acting via C8/C2=C4 2, 5
40 [40, 4] C5 \(\rtimes\) Q8  Dicyclic group 2, 5
40 [40, 5] C4\(\times\)D10  Direct product of C4 and D5 2, 5
40 [40, 6] D40  Dihedral group 2, 5
40 [40, 7] C2\(\times\)(C5  \(\rtimes\) C4) Direct product of C2 and Dic5 2, 5
40 [40, 8] (C10 \(\times\)C2)\(\rtimes\) C2  The semidirect product of C5 and D4 acting via D4/C22=C2 2, 5
40 [40, 9] C20\(\times\)C2  Abelian group of type [2, 20] 2, 5
40 [40, 10] C5\(\times\)D8  Direct product of C5 and D4 2, 5
40 [40, 11] C5\(\times\)Q8  Direct product of C5 and Q8 2, 5
40 [40, 12] C2\(\times\)(C5  \(\rtimes\) C4) Direct product of C2 and F5 2, 5
40 [40, 13] C2\(\times\)C2\(\times\)D10  Direct product of C22 and D5 2, 5
40 [40, 14] C10\(\times\)C2\(\times\)C2  Abelian group of type [2, 2,10] 2, 5
41 [41, 1] C41  Cyclic group 41
42 [42, 1] C7 \(\rtimes\) C6  Frobenius group 2, 3, 7
42 [42, 2] C2\(\times\)(C7  \(\rtimes\) C3) Direct product of C2 and C7\(\rtimes\)C3 2, 3, 7
42 [42, 3] C7\(\times\)S3  Direct product of C7 and S3 2, 3, 7
42 [42, 4] C3\(\times\)D14  Direct product of C3 and D7 2, 3, 7
42 [42, 5] D42  Dihedral group 2, 3, 7
42 [42, 6] C42  Cyclic group 2, 3, 7
43 [43, 1] C43  Cyclic group 43
44 [44, 1] C11 \(\rtimes\) C4  Dicyclic group 2, 11
44 [44, 2] C44  Cyclic group 2, 11
44 [44, 3] D44  Dihedral group 2, 11
44 [44, 4] C22\(\times\)C2  Abelian group of type [2, 22] 2, 11
45 [45, 1] C45  Cyclic group 3, 5
45 [45, 2] C15\(\times\)C3  Abelian group of type [3, 15] 3, 5
46 [46, 1] D46  Dihedral group 2, 23
46 [46, 2] C46  Cyclic group 2, 23
47 [47, 1] C47  Cyclic group 47
48 [48, 1] C3 \(\rtimes\) C16  The semidirect product of C3 and C16 acting via C16/C8=C2 2, 3
48 [48, 2] C48  Cyclic group 2, 3
48 [48, 3] (C4 \(\times\)C4)\(\rtimes\) C3  The semidirect product of C42 and C3 acting faithfully 2, 3
48 [48, 4] C8\(\times\)S3  Direct product of C8 and S3 2, 3
48 [48, 5] C24 \(\rtimes\) C2  3rd semidirect product of C8 and S3 acting via S3/C3=C2 2, 3
48 [48, 6] C24 \(\rtimes\) C2  2nd semidirect product of C24 and C2 acting faithfully 2, 3
48 [48, 7] D48  Dihedral group 2, 3
48 [48, 8] C3 \(\rtimes\) Q16  Dicyclic group 2, 3
48 [48, 9] C2\(\times\)(C3  \(\rtimes\) C8) Direct product of C2 and C3\(\rtimes\)C8 2, 3
48 [48, 10] (C3  \(\rtimes\) C8)\(\rtimes\) C2  The non-split extension by C4 of Dic3 acting via Dic3/C6=C2 2, 3
48 [48, 11] C4\(\times\)(C3  \(\rtimes\) C4) Direct product of C4 and Dic3 2, 3
48 [48, 12] (C3  \(\rtimes\) C4)\(\rtimes\) C4  The semidirect product of Dic3 and C4 acting via C4/C2=C2 2, 3
48 [48, 13] C12 \(\rtimes\) C4  The semidirect product of C4 and Dic3 acting via Dic3/C6=C2 2, 3
48 [48, 14] (C12 \(\times\)C2)\(\rtimes\) C2  The semidirect product of D6 and C4 acting via C4/C2=C2 2, 3
48 [48, 15] (C3 \(\times\)D8)\(\rtimes\) C2  The semidirect product of D4 and S3 acting via S3/C3=C2 2, 3
48 [48, 16] (C3  \(\rtimes\) Q8)\(\rtimes\) C2  The non-split extension by D4 of S3 acting via S3/C3=C2 2, 3
48 [48, 17] (C3 \(\times\)Q8)\(\rtimes\) C2  The semidirect product of Q8 and S3 acting via S3/C3=C2 2, 3
48 [48, 18] C3 \(\rtimes\) Q16  The semidirect product of C3 and Q16 acting via Q16/Q8=C2 2, 3
48 [48, 19] (C6 \(\times\)C2)\(\rtimes\) C4  7th non-split extension by C6 of D4 acting via D4/C22=C2 2, 3
48 [48, 20] C12\(\times\)C4  Abelian group of type [4, 12] 2, 3
48 [48, 21] C3\(\times\)((C4\(\times\)C2)\(\rtimes\) C2) Direct product of C3 and C22\(\rtimes\)C4 2, 3
48 [48, 22] C3\(\times\)(C4  \(\rtimes\) C4) Direct product of C3 and C4\(\rtimes\)C4 2, 3
48 [48, 23] C24\(\times\)C2  Abelian group of type [2, 24] 2, 3
48 [48, 24] C3\(\times\)(C8  \(\rtimes\) C2) Direct product of C3 and M4(2) 2, 3
48 [48, 25] C3\(\times\)D16  Direct product of C3 and D8 2, 3
48 [48, 26] C3\(\times\)QD16  Direct product of C3 and SD16 2, 3
48 [48, 27] C3\(\times\)Q16  Direct product of C3 and Q16 2, 3
48 [48, 28] C2.S4 = SL(2, 3).C2  Conformal special unitary group on 𝔽32 2, 3
48 [48, 29] GL2(𝔽3) General linear group on 𝔽32 2, 3
48 [48, 30] A4 \(\rtimes\) C4  The semidirect product of A4 and C4 acting via C4/C2=C2 2, 3
48 [48, 31] C4\(\times\)A4  Direct product of C4 and A4 2, 3
48 [48, 32] C2\(\times\)SL(2, 3)  Direct product of C2 and SL2(𝔽3) 2, 3
48 [48, 33] ((C4\(\times\)C2)\(\rtimes\) C2)\(\rtimes\) C3  The central extension by C4 of A4 2, 3
48 [48, 34] C2\(\times\)(C3  \(\rtimes\) Q8) Direct product of C2 and Dic6 2, 3
48 [48, 35] C2\(\times\)C4\(\times\)S3  Direct product of C2×C4 and S3 2, 3
48 [48, 36] C2\(\times\)D24  Direct product of C2 and D12 2, 3
48 [48, 37] (C12 \(\times\)C2)\(\rtimes\) C2  Central product of C4 and D12 2, 3
48 [48, 38] D8\(\times\)S3  Direct product of S3 and D4 2, 3
48 [48, 39] (C4 \(\times\)S3)\(\rtimes\) C2  The semidirect product of D4 and S3 acting through Inn(D4) 2, 3
48 [48, 40] Q8\(\times\)S3  Direct product of S3 and Q8 2, 3
48 [48, 41] (C4 \(\times\)S3)\(\rtimes\) C2  The semidirect product of Q8 and S3 acting through Inn(Q8) 2, 3
48 [48, 42] C2\(\times\)C2\(\times\)(C3  \(\rtimes\) C4) Direct product of C22 and Dic3 2, 3
48 [48, 43] C2\(\times\)((C6\(\times\)C2)\(\rtimes\) C2) Direct product of C2 and C3\(\rtimes\)D4 2, 3
48 [48, 44] C12\(\times\)C2\(\times\)C2  Abelian group of type [2, 2,12] 2, 3
48 [48, 45] C6\(\times\)D8  Direct product of C6 and D4 2, 3
48 [48, 46] C6\(\times\)Q8  Direct product of C6 and Q8 2, 3
48 [48, 47] C3\(\times\)((C4\(\times\)C2)\(\rtimes\) C2) Direct product of C3 and C4○D4 2, 3
48 [48, 48] C2\(\times\)S4  Direct product of C2 and S4 2, 3
48 [48, 49] C2\(\times\)C2\(\times\)A4  Direct product of C22 and A4 2, 3
48 [48, 50] (C2 \(\times\)C2\(\times\)C2\(\times\)C2)\(\rtimes\) C3  The semidirect product of C22 and A4 acting via A4/C22=C3 2, 3
48 [48, 51] C2\(\times\)C2\(\times\)C2\(\times\)S3  Direct product of C23 and S3 2, 3
48 [48, 52] C6\(\times\)C2\(\times\)C2\(\times\)C2  Abelian group of type [2, 2,2, 6] 2, 3
49 [49, 1] C49  Cyclic group 7
49 [49, 2] C7\(\times\)C7  Elementary abelian group of type [7, 7] 7
50 [50, 1] D50  Dihedral group 2, 5
50 [50, 2] C50  Cyclic group 2, 5
50 [50, 3] C5\(\times\)D10  Direct product of C5 and D5 2, 5
50 [50, 4] (C5 \(\times\)C5)\(\rtimes\) C2  The semidirect product of C5 and D5 acting via D5/C5=C2 2, 5
50 [50, 5] C10\(\times\)C5  Abelian group of type [5, 10] 2, 5
51 [51, 1] C51  Cyclic group 3, 17
52 [52, 1] C13 \(\rtimes\) C4  Dicyclic group 2, 13
52 [52, 2] C52  Cyclic group 2, 13
52 [52, 3] C13 \(\rtimes\) C4  The semidirect product of C13 and C4 acting faithfully 2, 13
52 [52, 4] D52  Dihedral group 2, 13
52 [52, 5] C26\(\times\)C2  Abelian group of type [2, 26] 2, 13
53 [53, 1] C53  Cyclic group 53
54 [54, 1] D54  Dihedral group 2, 3
54 [54, 2] C54  Cyclic group 2, 3
54 [54, 3] C3\(\times\)D18  Direct product of C3 and D9 2, 3
54 [54, 4] C9\(\times\)S3  Direct product of C9 and S3 2, 3
54 [54, 5] (C3 \(\times\)C3)\(\rtimes\) C6  The semidirect product of C32 and C6 acting faithfully 2, 3
54 [54, 6] C9 \(\rtimes\) C6  The semidirect product of C9 and C6 acting faithfully 2, 3
54 [54, 7] (C9 \(\times\)C3)\(\rtimes\) C2  The semidirect product of C9 and S3 acting via S3/C3=C2 2, 3
54 [54, 8] ((C3\(\times\)C3)\(\rtimes\) C3)\(\rtimes\) C2  2nd semidirect product of He3 and C2 acting faithfully 2, 3
54 [54, 9] C18\(\times\)C3  Abelian group of type [3, 18] 2, 3
54 [54, 10] C2\(\times\)((C3\(\times\)C3)\(\rtimes\) C3) Direct product of C2 and He3 2, 3
54 [54, 11] C2\(\times\)(C9  \(\rtimes\) C3) Direct product of C2 and 31+2 2, 3
54 [54, 12] C3\(\times\)C3\(\times\)S3  Direct product of C32 and S3 2, 3
54 [54, 13] C3\(\times\)((C3\(\times\)C3)\(\rtimes\) C2) Direct product of C3 and C3\(\rtimes\)S3 2, 3
54 [54, 14] (C3 \(\times\)C3\(\times\)C3)\(\rtimes\) C2  3rd semidirect product of C33 and C2 acting faithfully 2, 3
54 [54, 15] C6\(\times\)C3\(\times\)C3  Abelian group of type [3, 3,6] 2, 3
55 [55, 1] C11 \(\rtimes\) C5  The semidirect product of C11 and C5 acting faithfully 5, 11
55 [55, 2] C55  Cyclic group 5, 11
56 [56, 1] C7 \(\rtimes\) C8  The semidirect product of C7 and C8 acting via C8/C4=C2 2, 7
56 [56, 2] C56  Cyclic group 2, 7
56 [56, 3] C7 \(\rtimes\) Q8  Dicyclic group 2, 7
56 [56, 4] C4\(\times\)D14  Direct product of C4 and D7 2, 7
56 [56, 5] D56  Dihedral group 2, 7
56 [56, 6] C2\(\times\)(C7  \(\rtimes\) C4) Direct product of C2 and Dic7 2, 7
56 [56, 7] (C14 \(\times\)C2)\(\rtimes\) C2  The semidirect product of C7 and D4 acting via D4/C22=C2 2, 7
56 [56, 8] C28\(\times\)C2  Abelian group of type [2, 28] 2, 7
56 [56, 9] C7\(\times\)D8  Direct product of C7 and D4 2, 7
56 [56, 10] C7\(\times\)Q8  Direct product of C7 and Q8 2, 7
56 [56, 11] (C2 \(\times\)C2\(\times\)C2)\(\rtimes\) C7  Frobenius group 2, 7
56 [56, 12] C2\(\times\)C2\(\times\)D14  Direct product of C22 and D7 2, 7
56 [56, 13] C14\(\times\)C2\(\times\)C2  Abelian group of type [2, 2,14] 2, 7
57 [57, 1] C19 \(\rtimes\) C3  The semidirect product of C19 and C3 acting faithfully 3, 19
57 [57, 2] C57  Cyclic group 3, 19
58 [58, 1] D58  Dihedral group 2, 29
58 [58, 2] C58  Cyclic group 2, 29
59 [59, 1] C59  Cyclic group 59
60 [60, 1] C5\(\times\)(C3  \(\rtimes\) C4) Direct product of C5 and Dic3 2, 3, 5
60 [60, 2] C3\(\times\)(C5  \(\rtimes\) C4) Direct product of C3 and Dic5 2, 3, 5
60 [60, 3] C15 \(\rtimes\) C4  Dicyclic group 2, 3, 5
60 [60, 4] C60  Cyclic group 2, 3, 5
60 [60, 5] A5  Alternating group on 5 letters 2, 3, 5
60 [60, 6] C3\(\times\)(C5  \(\rtimes\) C4) Direct product of C3 and F5 2, 3, 5
60 [60, 7] C15 \(\rtimes\) C4  The semidirect product of C3 and F5 acting via F5/D5=C2 2, 3, 5
60 [60, 8] S3\(\times\)D10  Direct product of S3 and D5 2, 3, 5
60 [60, 9] C5\(\times\)A4  Direct product of C5 and A4 2, 3, 5
60 [60, 10] C6\(\times\)D10  Direct product of C6 and D5 2, 3, 5
60 [60, 11] C10\(\times\)S3  Direct product of C10 and S3 2, 3, 5
60 [60, 12] D60  Dihedral group 2, 3, 5
60 [60, 13] C30\(\times\)C2  Abelian group of type [2, 30] 2, 3, 5
61 [61, 1] C61 Cyclic group 61
62 [62, 1] D31 Dihedral group 2, 31
62 [62, 2] C62 Cyclic group 2, 31
63 [63, 1] C7:C9 The semidirect product of C7 and C9 acting via C9/C3=C3 3, 7
63 [63, 2] C63 Cyclic group 3, 7
63 [63, 3] C3\(\times\)C7:C3 Direct product of C3 and C7⋊C3 3, 7
63 [63, 4] C3\(\times\)C21 Abelian group of type [3, 21] 3, 7
64 [64, 1] C64 Cyclic group 2
64 [64, 2] C82 Abelian group of type [8, 8] 2
64 [64, 3] C8:C8 3rd semidirect product of C8 and C8 acting via C8/C4=C2 2
64 [64, 4] C23:C8 The semidirect product of C23 and C8 acting via C8/C2=C4 2
64 [64, 5] C22.M4(2) 2nd non-split extension by C22 of M4(2) acting via M4(2)/C2×C4=C2 2
64 [64, 6] D4:C8 The semidirect product of D4 and C8 acting via C8/C4=C2 2
64 [64, 7] Q8:C8 The semidirect product of Q8 and C8 acting via C8/C4=C2 2
64 [64, 8] C22.SD16 1st non-split extension by C22 of SD16 acting via SD16/Q8=C2 2
64 [64, 9] C23.31D4 2nd non-split extension by C23 of D4 acting via D4/C22=C2 2
64 [64, 10] C42.C22 1st non-split extension by C42 of C22 acting faithfully 2
64 [64, 11] C42.2C22 2nd non-split extension by C42 of C22 acting faithfully 2
64 [64, 12] C4.D8 1st non-split extension by C4 of D8 acting via D8/D4=C2 2
64 [64, 13] C4.10D8 2nd non-split extension by C4 of D8 acting via D8/D4=C2 2
64 [64, 14] C4.6Q16 2nd non-split extension by C4 of Q16 acting via Q16/Q8=C2 2
64 [64, 15] C8:2C8 2nd semidirect product of C8 and C8 acting via C8/C4=C2 2
64 [64, 16] C8:1C8 1st semidirect product of C8 and C8 acting via C8/C4=C2 2
64 [64, 17] C22.7C42 2nd central extension by C22 of C42 2
64 [64, 18] C4.9C42 1st central stem extension by C4 of C42 2
64 [64, 19] C4.10C42 2nd central stem extension by C4 of C42 2
64 [64, 20] C42:6C4 3rd semidirect product of C42 and C4 acting via C4/C2=C2 2
64 [64, 21] C22.4Q16 1st central extension by C22 of Q16 2
64 [64, 22] C4.C42 3rd non-split extension by C4 of C42 acting via C42/C2×C4=C2 2
64 [64, 23] C23.9D4 2nd non-split extension by C23 of D4 acting via D4/C2=C22 2
64 [64, 24] C22.C42 2nd non-split extension by C22 of C42 acting via C42/C2×C4=C2 2
64 [64, 25] M4(2):4C4 4th semidirect product of M4(2) and C4 acting via C4/C2=C2 2
64 [64, 26] C4\(\times\)C16 Abelian group of type [4, 16] 2
64 [64, 27] C16:5C4 3rd semidirect product of C16 and C4 acting via C4/C2=C2 2
64 [64, 28] C16:C4 2nd semidirect product of C16 and C4 acting faithfully 2
64 [64, 29] C22:C16 The semidirect product of C22 and C16 acting via C16/C8=C2 2
64 [64, 30] C23.C8 The non-split extension by C23 of C8 acting via C8/C2=C4 2
64 [64, 31] D4.C8 The non-split extension by D4 of C8 acting via C8/C4=C2 2
64 [64, 32] C2wrC4 Wreath product of C2 by C4 2
64 [64, 33] C23.D4 2nd non-split extension by C23 of D4 acting faithfully 2
64 [64, 34] C42:C4 2nd semidirect product of C42 and C4 acting faithfully 2
64 [64, 35] C42:3C4 3rd semidirect product of C42 and C4 acting faithfully 2
64 [64, 36] C42.C4 2nd non-split extension by C42 of C4 acting faithfully 2
64 [64, 37] C42.3C4 3rd non-split extension by C42 of C4 acting faithfully 2
64 [64, 38] C2.D16 1st central extension by C2 of D16 2
64 [64, 39] C2.Q32 1st central extension by C2 of Q32 2
64 [64, 40] D8.C4 1st non-split extension by D8 of C4 acting via C4/C2=C2 2
64 [64, 41] D8:2C4 2nd semidirect product of D8 and C4 acting via C4/C2=C2 2
64 [64, 42] M5(2):C2 6th semidirect product of M5(2) and C2 acting faithfully 2
64 [64, 43] C8.17D4 4th non-split extension by C8 of D4 acting via D4/C22=C2 2
64 [64, 44] C4:C16 The semidirect product of C4 and C16 acting via C16/C8=C2 2
64 [64, 45] C8.C8 1st non-split extension by C8 of C8 acting via C8/C4=C2 2
64 [64, 46] C8.Q8 The non-split extension by C8 of Q8 acting via Q8/C2=C22 2
64 [64, 47] C16:3C4 1st semidirect product of C16 and C4 acting via C4/C2=C2 2
64 [64, 48] C16:4C4 2nd semidirect product of C16 and C4 acting via C4/C2=C2 2
64 [64, 49] C8.4Q8 3rd non-split extension by C8 of Q8 acting via Q8/C4=C2 2
64 [64, 50] C2\(\times\)C32 Abelian group of type [2, 32] 2
64 [64, 51] M6(2) Modular maximal-cyclic group 2
64 [64, 52] D32 Dihedral group 2
64 [64, 53] SD64 Semidihedral group 2
64 [64, 54] Q64 Generalised quaternion group 2
64 [64, 55] C43 Abelian group of type [4, 4,4] 2
64 [64, 56] C2\(\times\)C2.C42 Direct product of C2 and C2.C42 2
64 [64, 57] C42:4C4 1st semidirect product of C42 and C4 acting via C4/C2=C2 2
64 [64, 58] C4\(\times\)C22:C4 Direct product of C4 and C22⋊C4 2
64 [64, 59] C4\(\times\)C4:C4 Direct product of C4 and C4⋊C4 2
64 [64, 60] C24:3C4 1st semidirect product of C24 and C4 acting via C4/C2=C2 2
64 [64, 61] C23.7Q8 2nd non-split extension by C23 of Q8 acting via Q8/C4=C2 2
64 [64, 62] C23.34D4 5th non-split extension by C23 of D4 acting via D4/C22=C2 2
64 [64, 63] C42:8C4 5th semidirect product of C42 and C4 acting via C4/C2=C2 2
64 [64, 64] C42:5C4 2nd semidirect product of C42 and C4 acting via C4/C2=C2 2
64 [64, 65] C42:9C4 6th semidirect product of C42 and C4 acting via C4/C2=C2 2
64 [64, 66] C23.8Q8 3rd non-split extension by C23 of Q8 acting via Q8/C4=C2 2
64 [64, 67] C23.23D4 2nd non-split extension by C23 of D4 acting via D4/C4=C2 2
64 [64, 68] C23.63C23 13rd central extension by C23 of C23 2
64 [64, 69] C24.C22 2nd non-split extension by C24 of C22 acting faithfully 2
64 [64, 70] C23.65C23 15th central extension by C23 of C23 2
64 [64, 71] C24.3C22 3rd non-split extension by C24 of C22 acting faithfully 2
64 [64, 72] C23.67C23 17th central extension by C23 of C23 2
64 [64, 73] C23:2D4 1st semidirect product of C23 and D4 acting via D4/C2=C22 2
64 [64, 74] C23:Q8 1st semidirect product of C23 and Q8 acting via Q8/C2=C22 2
64 [64, 75] C23.10D4 3rd non-split extension by C23 of D4 acting via D4/C2=C22 2
64 [64, 76] C23.78C23 4th central stem extension by C23 of C23 2
64 [64, 77] C23.Q8 3rd non-split extension by C23 of Q8 acting via Q8/C2=C22 2
64 [64, 78] C23.11D4 4th non-split extension by C23 of D4 acting via D4/C2=C22 2
64 [64, 79] C23.81C23 7th central stem extension by C23 of C23 2
64 [64, 80] C23.4Q8 4th non-split extension by C23 of Q8 acting via Q8/C2=C22 2
64 [64, 81] C23.83C23 9th central stem extension by C23 of C23 2
64 [64, 82] C23.84C23 10th central stem extension by C23 of C23 2
64 [64, 83] C2\(\times\)C4\(\times\)C8 Abelian group of type [2, 4,8] 2
64 [64, 84] C2\(\times\)C8:C4 Direct product of C2 and C8⋊C4 2
64 [64, 85] C4\(\times\)M4(2) Direct product of C4 and M4(2) 2
64 [64, 86] C8o2M4(2) Central product of C8 and M4(2) 2
64 [64, 87] C2\(\times\)C22:C8 Direct product of C2 and C22⋊C8 2
64 [64, 88] C24.4C4 2nd non-split extension by C24 of C4 acting via C4/C2=C2 2
64 [64, 89] (C22\(\times\)C8):C2 2nd semidirect product of C22×C8 and C2 acting faithfully 2
64 [64, 90] C2\(\times\)C23:C4 Direct product of C2 and C23⋊C4 2
64 [64, 91] C23.C23 2nd non-split extension by C23 of C23 acting via C23/C2=C22 2
64 [64, 92] C2\(\times\)C4.D4 Direct product of C2 and C4.D4 2
64 [64, 93] C2\(\times\)C4.10D4 Direct product of C2 and C4.10D4 2
64 [64, 94] M4(2).8C22 3rd non-split extension by M4(2) of C22 acting via C22/C2=C2 2
64 [64, 95] C2\(\times\)D4:C4 Direct product of C2 and D4⋊C4 2
64 [64, 96] C2\(\times\)Q8:C4 Direct product of C2 and Q8⋊C4 2
64 [64, 97] C23.24D4 3rd non-split extension by C23 of D4 acting via D4/C4=C2 2
64 [64, 98] C23.36D4 7th non-split extension by C23 of D4 acting via D4/C22=C2 2
64 [64, 99] C23.37D4 8th non-split extension by C23 of D4 acting via D4/C22=C2 2
64 [64, 100] C23.38D4 9th non-split extension by C23 of D4 acting via D4/C22=C2 2
64 [64, 101] C2\(\times\)C4wrC2 Direct product of C2 and C4≀C2 2
64 [64, 102] C42:C22 1st semidirect product of C42 and C22 acting faithfully 2
64 [64, 103] C2\(\times\)C4:C8 Direct product of C2 and C4⋊C8 2
64 [64, 104] C4:M4(2) The semidirect product of C4 and M4(2) acting via M4(2)/C2×C4=C2 2
64 [64, 105] C42.6C22 6th non-split extension by C42 of C22 acting faithfully 2
64 [64, 106] C2\(\times\)C4.Q8 Direct product of C2 and C4.Q8 2
64 [64, 107] C2\(\times\)C2.D8 Direct product of C2 and C2.D8 2
64 [64, 108] C23.25D4 4th non-split extension by C23 of D4 acting via D4/C4=C2 2
64 [64, 109] M4(2):C4 1st semidirect product of M4(2) and C4 acting via C4/C2=C2 2
64 [64, 110] C2\(\times\)C8.C4 Direct product of C2 and C8.C4 2
64 [64, 111] M4(2).C4 1st non-split extension by M4(2) of C4 acting via C4/C2=C2 2
64 [64, 112] C42.12C4 9th non-split extension by C42 of C4 acting via C4/C2=C2 2
64 [64, 113] C42.6C4 3rd non-split extension by C42 of C4 acting via C4/C2=C2 2
64 [64, 114] C42.7C22 7th non-split extension by C42 of C22 acting faithfully 2
64 [64, 115] C8\(\times\)D4 Direct product of C8 and D4 2
64 [64, 116] C8:9D4 3rd semidirect product of C8 and D4 acting via D4/C22=C2 2
64 [64, 117] C8:6D4 3rd semidirect product of C8 and D4 acting via D4/C4=C2 2
64 [64, 118] C4\(\times\)D8 Direct product of C4 and D8 2
64 [64, 119] C4\(\times\)SD16 Direct product of C4 and SD16 2
64 [64, 120] C4\(\times\)Q16 Direct product of C4 and Q16 2
64 [64, 121] SD16:C4 1st semidirect product of SD16 and C4 acting via C4/C2=C2 2
64 [64, 122] Q16:C4 3rd semidirect product of Q16 and C4 acting via C4/C2=C2 2
64 [64, 123] D8:C4 3rd semidirect product of D8 and C4 acting via C4/C2=C2 2
64 [64, 124] C8oD8 Central product of C8 and D8 2
64 [64, 125] C8.26D4 13rd non-split extension by C8 of D4 acting via D4/C22=C2 2
64 [64, 126] C8\(\times\)Q8 Direct product of C8 and Q8 2
64 [64, 127] C8:4Q8 3rd semidirect product of C8 and Q8 acting via Q8/C4=C2 2
64 [64, 128] C22:D8 The semidirect product of C22 and D8 acting via D8/D4=C2 2
64 [64, 129] Q8:D4 1st semidirect product of Q8 and D4 acting via D4/C22=C2 2
64 [64, 130] D4:D4 2nd semidirect product of D4 and D4 acting via D4/C22=C2 2
64 [64, 131] C22:SD16 The semidirect product of C22 and SD16 acting via SD16/D4=C2 2
64 [64, 132] C22:Q16 The semidirect product of C22 and Q16 acting via Q16/Q8=C2 2
64 [64, 133] D4.7D4 2nd non-split extension by D4 of D4 acting via D4/C22=C2 2
64 [64, 134] D4:4D4 3rd semidirect product of D4 and D4 acting via D4/C22=C2 2
64 [64, 135] D4.8D4 3rd non-split extension by D4 of D4 acting via D4/C22=C2 2
64 [64, 136] D4.9D4 4th non-split extension by D4 of D4 acting via D4/C22=C2 2
64 [64, 137] D4.10D4 5th non-split extension by D4 of D4 acting via D4/C22=C2 2
64 [64, 138] C2wrC22 Wreath product of C2 by C22 2
64 [64, 139] C23.7D4 7th non-split extension by C23 of D4 acting faithfully 2
64 [64, 140] C4:D8 The semidirect product of C4 and D8 acting via D8/D4=C2 2
64 [64, 141] C4:SD16 The semidirect product of C4 and SD16 acting via SD16/Q8=C2 2
64 [64, 142] D4.D4 1st non-split extension by D4 of D4 acting via D4/C4=C2 2
64 [64, 143] C4:2Q16 The semidirect product of C4 and Q16 acting via Q16/Q8=C2 2
64 [64, 144] D4.2D4 2nd non-split extension by D4 of D4 acting via D4/C4=C2 2
64 [64, 145] Q8.D4 2nd non-split extension by Q8 of D4 acting via D4/C4=C2 2
64 [64, 146] C8:8D4 2nd semidirect product of C8 and D4 acting via D4/C22=C2 2
64 [64, 147] C8:7D4 1st semidirect product of C8 and D4 acting via D4/C22=C2 2
64 [64, 148] C8.18D4 5th non-split extension by C8 of D4 acting via D4/C22=C2 2
64 [64, 149] C8:D4 1st semidirect product of C8 and D4 acting via D4/C2=C22 2
64 [64, 150] C8:2D4 2nd semidirect product of C8 and D4 acting via D4/C2=C22 2
64 [64, 151] C8.D4 1st non-split extension by C8 of D4 acting via D4/C2=C22 2
64 [64, 152] D4.3D4 3rd non-split extension by D4 of D4 acting via D4/C4=C2 2
64 [64, 153] D4.4D4 4th non-split extension by D4 of D4 acting via D4/C4=C2 2
64 [64, 154] D4.5D4 5th non-split extension by D4 of D4 acting via D4/C4=C2 2
64 [64, 155] D4:Q8 1st semidirect product of D4 and Q8 acting via Q8/C4=C2 2
64 [64, 156] Q8:Q8 1st semidirect product of Q8 and Q8 acting via Q8/C4=C2 2
64 [64, 157] D4:2Q8 2nd semidirect product of D4 and Q8 acting via Q8/C4=C2 2
64 [64, 158] C4.Q16 3rd non-split extension by C4 of Q16 acting via Q16/Q8=C2 2
64 [64, 159] D4.Q8 The non-split extension by D4 of Q8 acting via Q8/C4=C2 2
64 [64, 160] Q8.Q8 The non-split extension by Q8 of Q8 acting via Q8/C4=C2 2
64 [64, 161] C22.D8 3rd non-split extension by C22 of D8 acting via D8/D4=C2 2
64 [64, 162] C23.46D4 17th non-split extension by C23 of D4 acting via D4/C22=C2 2
64 [64, 163] C23.19D4 12nd non-split extension by C23 of D4 acting via D4/C2=C22 2
64 [64, 164] C23.47D4 18th non-split extension by C23 of D4 acting via D4/C22=C2 2
64 [64, 165] C23.48D4 19th non-split extension by C23 of D4 acting via D4/C22=C2 2
64 [64, 166] C23.20D4 13rd non-split extension by C23 of D4 acting via D4/C2=C22 2
64 [64, 167] C4.4D8 4th non-split extension by C4 of D8 acting via D8/C8=C2 2
64 [64, 168] C4.SD16 4th non-split extension by C4 of SD16 acting via SD16/C8=C2 2
64 [64, 169] C42.78C22 21st non-split extension by C42 of C22 acting via C22/C2=C2 2
64 [64, 170] C42.28C22 28th non-split extension by C42 of C22 acting faithfully 2
64 [64, 171] C42.29C22 29th non-split extension by C42 of C22 acting faithfully 2
64 [64, 172] C42.30C22 30th non-split extension by C42 of C22 acting faithfully 2
64 [64, 173] C8:5D4 2nd semidirect product of C8 and D4 acting via D4/C4=C2 2
64 [64, 174] C8:4D4 1st semidirect product of C8 and D4 acting via D4/C4=C2 2
64 [64, 175] C4:Q16 The semidirect product of C4 and Q16 acting via Q16/C8=C2 2
64 [64, 176] C8.12D4 8th non-split extension by C8 of D4 acting via D4/C4=C2 2
64 [64, 177] C8:3D4 3rd semidirect product of C8 and D4 acting via D4/C2=C22 2
64 [64, 178] C8.2D4 2nd non-split extension by C8 of D4 acting via D4/C2=C22 2
64 [64, 179] C8:3Q8 2nd semidirect product of C8 and Q8 acting via Q8/C4=C2 2
64 [64, 180] C8.5Q8 4th non-split extension by C8 of Q8 acting via Q8/C4=C2 2
64 [64, 181] C8:2Q8 1st semidirect product of C8 and Q8 acting via Q8/C4=C2 2
64 [64, 182] C8:Q8 The semidirect product of C8 and Q8 acting via Q8/C2=C22 2
64 [64, 183] C22\(\times\)C16 Abelian group of type [2, 2,16] 2
64 [64, 184] C2\(\times\)M5(2) Direct product of C2 and M5(2) 2
64 [64, 185] D4oC16 Central product of D4 and C16 2
64 [64, 186] C2\(\times\)D16 Direct product of C2 and D16 2
64 [64, 187] C2\(\times\)SD32 Direct product of C2 and SD32 2
64 [64, 188] C2\(\times\)Q32 Direct product of C2 and Q32 2
64 [64, 189] C4oD16 Central product of C4 and D16 2
64 [64, 190] C16:C22 The semidirect product of C16 and C22 acting faithfully 2
64 [64, 191] Q32:C2 2nd semidirect product of Q32 and C2 acting faithfully 2
64 [64, 192] C22\(\times\)C42 Abelian group of type [2, 2,4, 4] 2
64 [64, 193] C22\(\times\)C22:C4 Direct product of C22 and C22⋊C4 2
64 [64, 194] C22\(\times\)C4:C4 Direct product of C22 and C4⋊C4 2
64 [64, 195] C2\(\times\)C42:C2 Direct product of C2 and C42⋊C2 2
64 [64, 196] C2\(\times\)C4\(\times\)D4 Direct product of C2×C4 and D4 2
64 [64, 197] C2\(\times\)C4\(\times\)Q8 Direct product of C2×C4 and Q8 2
64 [64, 198] C4\(\times\)C4oD4 Direct product of C4 and C4○D4 2
64 [64, 199] C22.11C24 7th central extension by C22 of C24 2
64 [64, 200] C23.32C23 5th non-split extension by C23 of C23 acting via C23/C22=C2 2
64 [64, 201] C23.33C23 6th non-split extension by C23 of C23 acting via C23/C22=C2 2
64 [64, 202] C2\(\times\)C22wrC2 Direct product of C2 and C22≀C2 2
64 [64, 203] C2\(\times\)C4:D4 Direct product of C2 and C4⋊D4 2
64 [64, 204] C2\(\times\)C22:Q8 Direct product of C2 and C22⋊Q8 2
64 [64, 205] C2\(\times\)C22.D4 Direct product of C2 and C22.D4 2
64 [64, 206] C22.19C24 5th central stem extension by C22 of C24 2
64 [64, 207] C2\(\times\)C4.4D4 Direct product of C2 and C4.4D4 2
64 [64, 208] C2\(\times\)C42.C2 Direct product of C2 and C42.C2 2
64 [64, 209] C2\(\times\)C42:2C2 Direct product of C2 and C42⋊2C2 2
64 [64, 210] C23.36C23 9th non-split extension by C23 of C23 acting via C23/C22=C2 2
64 [64, 211] C2\(\times\)C4:1D4 Direct product of C2 and C4⋊1D4 2
64 [64, 212] C2\(\times\)C4:Q8 Direct product of C2 and C4⋊Q8 2
64 [64, 213] C22.26C24 12nd central stem extension by C22 of C24 2
64 [64, 214] C23.37C23 10th non-split extension by C23 of C23 acting via C23/C22=C2 2
64 [64, 215] C23:3D4 2nd semidirect product of C23 and D4 acting via D4/C2=C22 2
64 [64, 216] C22.29C24 15th central stem extension by C22 of C24 2
64 [64, 217] C23.38C23 11st non-split extension by C23 of C23 acting via C23/C22=C2 2
64 [64, 218] C22.31C24 17th central stem extension by C22 of C24 2
64 [64, 219] C22.32C24 18th central stem extension by C22 of C24 2
64 [64, 220] C22.33C24 19th central stem extension by C22 of C24 2
64 [64, 221] C22.34C24 20th central stem extension by C22 of C24 2
64 [64, 222] C22.35C24 21st central stem extension by C22 of C24 2
64 [64, 223] C22.36C24 22nd central stem extension by C22 of C24 2
64 [64, 224] C23:2Q8 2nd semidirect product of C23 and Q8 acting via Q8/C2=C22 2
64 [64, 225] C23.41C23 14th non-split extension by C23 of C23 acting via C23/C22=C2 2
64 [64, 226] D42 Direct product of D4 and D4 2
64 [64, 227] D4:5D4 1st semidirect product of D4 and D4 acting through Inn(D4) 2
64 [64, 228] D4:6D4 2nd semidirect product of D4 and D4 acting through Inn(D4) 2
64 [64, 229] Q8:5D4 1st semidirect product of Q8 and D4 acting through Inn(Q8) 2
64 [64, 230] D4\(\times\)Q8 Direct product of D4 and Q8 2
64 [64, 231] Q8:6D4 2nd semidirect product of Q8 and D4 acting through Inn(Q8) 2
64 [64, 232] C22.45C24 31st central stem extension by C22 of C24 2
64 [64, 233] C22.46C24 32nd central stem extension by C22 of C24 2
64 [64, 234] C22.47C24 33rd central stem extension by C22 of C24 2
64 [64, 235] D4:3Q8 The semidirect product of D4 and Q8 acting through Inn(D4) 2
64 [64, 236] C22.49C24 35th central stem extension by C22 of C24 2
64 [64, 237] C22.50C24 36th central stem extension by C22 of C24 2
64 [64, 238] Q8:3Q8 The semidirect product of Q8 and Q8 acting through Inn(Q8) 2
64 [64, 239] Q82 Direct product of Q8 and Q8 2
64 [64, 240] C22.53C24 39th central stem extension by C22 of C24 2
64 [64, 241] C22.54C24 40th central stem extension by C22 of C24 2
64 [64, 242] C24:C22 4th semidirect product of C24 and C22 acting faithfully 2
64 [64, 243] C22.56C24 42nd central stem extension by C22 of C24 2
64 [64, 244] C22.57C24 43rd central stem extension by C22 of C24 2
64 [64, 245] C22.58C24 44th central stem extension by C22 of C24 2
64 [64, 246] C23\(\times\)C8 Abelian group of type [2, 2,2, 8] 2
64 [64, 247] C22\(\times\)M4(2) Direct product of C22 and M4(2) 2
64 [64, 248] C2\(\times\)C8oD4 Direct product of C2 and C8○D4 2
64 [64, 249] Q8oM4(2) Central product of Q8 and M4(2) 2
64 [64, 250] C22\(\times\)D8 Direct product of C22 and D8 2
64 [64, 251] C22\(\times\)SD16 Direct product of C22 and SD16 2
64 [64, 252] C22\(\times\)Q16 Direct product of C22 and Q16 2
64 [64, 253] C2\(\times\)C4oD8 Direct product of C2 and C4○D8 2
64 [64, 254] C2\(\times\)C8:C22 Direct product of C2 and C8⋊C22 2
64 [64, 255] C2\(\times\)C8.C22 Direct product of C2 and C8.C22 2
64 [64, 256] D8:C22 4th semidirect product of D8 and C22 acting via C22/C2=C2 2
64 [64, 257] D4oD8 Central product of D4 and D8 2
64 [64, 258] D4oSD16 Central product of D4 and SD16 2
64 [64, 259] Q8oD8 Central product of Q8 and D8 2
64 [64, 260] C24\(\times\)C4 Abelian group of type [2, 2,2, 2,4] 2
64 [64, 261] D4\(\times\)C23 Direct product of C23 and D4 2
64 [64, 262] Q8\(\times\)C23 Direct product of C23 and Q8 2
64 [64, 263] C22\(\times\)C4oD4 Direct product of C22 and C4○D4 2
64 [64, 264] C2\(\times\)ES+(2, 2) Direct product of C2 and 2+ 1+4 2
64 [64, 265] C2\(\times\)ES-(2, 2) Direct product of C2 and 2- 1+4 2
64 [64, 266] C2.C25 6th central stem extension by C2 of C25 2
64 [64, 267] C26 Elementary abelian group of type [2, 2,2, 2,2, 2] 2
65 [65, 1] C65 Cyclic group 5, 13
66 [66, 1] S3\(\times\)C11 Direct product of C11 and S3 2, 3, 11
66 [66, 2] C3\(\times\)D11 Direct product of C3 and D11 2, 3, 11
66 [66, 3] D33 Dihedral group 2, 3, 11
66 [66, 4] C66 Cyclic group 2, 3, 11
67 [67, 1] C67 Cyclic group 67
68 [68, 1] Dic17 Dicyclic group 2, 17
68 [68, 2] C68 Cyclic group 2, 17
68 [68, 3] C17:C4 The semidirect product of C17 and C4 acting faithfully 2, 17
68 [68, 4] D34 Dihedral group 2, 17
68 [68, 5] C2\(\times\)C34 Abelian group of type [2, 34] 2, 17
69 [69, 1] C69 Cyclic group 3, 23
70 [70, 1] C7\(\times\)D5 Direct product of C7 and D5 2, 5, 7
70 [70, 2] C5\(\times\)D7 Direct product of C5 and D7 2, 5, 7
70 [70, 3] D35 Dihedral group 2, 5, 7
70 [70, 4] C70 Cyclic group 2, 5, 7
71 [71, 1] C71 Cyclic group 71
72 [72, 1] C9:C8 The semidirect product of C9 and C8 acting via C8/C4=C2 2, 3
72 [72, 2] C72 Cyclic group 2, 3
72 [72, 3] Q8:C9 The semidirect product of Q8 and C9 acting via C9/C3=C3 2, 3
72 [72, 4] Dic18 Dicyclic group 2, 3
72 [72, 5] C4\(\times\)D9 Direct product of C4 and D9 2, 3
72 [72, 6] D36 Dihedral group 2, 3
72 [72, 7] C2\(\times\)Dic9 Direct product of C2 and Dic9 2, 3
72 [72, 8] C9:D4 The semidirect product of C9 and D4 acting via D4/C22=C2 2, 3
72 [72, 9] C2\(\times\)C36 Abelian group of type [2, 36] 2, 3
72 [72, 10] D4\(\times\)C9 Direct product of C9 and D4 2, 3
72 [72, 11] Q8\(\times\)C9 Direct product of C9 and Q8 2, 3
72 [72, 12] C3\(\times\)C3:C8 Direct product of C3 and C3⋊C8 2, 3
72 [72, 13] C32:4C8 2nd semidirect product of C32 and C8 acting via C8/C4=C2 2, 3
72 [72, 14] C3\(\times\)C24 Abelian group of type [3, 24] 2, 3
72 [72, 15] C3.S4 The non-split extension by C3 of S4 acting via S4/A4=C2 2, 3
72 [72, 16] C2\(\times\)C3.A4 Direct product of C2 and C3.A4 2, 3
72 [72, 17] C22\(\times\)D9 Direct product of C22 and D9 2, 3
72 [72, 18] C22\(\times\)C18 Abelian group of type [2, 2,18] 2, 3
72 [72, 19] C32:2C8 The semidirect product of C32 and C8 acting via C8/C2=C4 2, 3
72 [72, 20] S3\(\times\)Dic3 Direct product of S3 and Dic3 2, 3
72 [72, 21] C6.D6 2nd non-split extension by C6 of D6 acting via D6/S3=C2 2, 3
72 [72, 22] D6:S3 1st semidirect product of D6 and S3 acting via S3/C3=C2 2, 3
72 [72, 23] C3:D12 The semidirect product of C3 and D12 acting via D12/D6=C2 2, 3
72 [72, 24] C32:2Q8 The semidirect product of C32 and Q8 acting via Q8/C2=C22 2, 3
72 [72, 25] C3\(\times\)SL(2, 3) Direct product of C3 and SL2(𝔽3) 2, 3
72 [72, 26] C3\(\times\)Dic6 Direct product of C3 and Dic6 2, 3
72 [72, 27] S3\(\times\)C12 Direct product of C12 and S3 2, 3
72 [72, 28] C3\(\times\)D12 Direct product of C3 and D12 2, 3
72 [72, 29] C6\(\times\)Dic3 Direct product of C6 and Dic3 2, 3
72 [72, 30] C3\(\times\)C3:D4 Direct product of C3 and C3⋊D4 2, 3
72 [72, 31] C32:4Q8 2nd semidirect product of C32 and Q8 acting via Q8/C4=C2 2, 3
72 [72, 32] C4\(\times\)C3:S3 Direct product of C4 and C3⋊S3 2, 3
72 [72, 33] C12:S3 1st semidirect product of C12 and S3 acting via S3/C3=C2 2, 3
72 [72, 34] C2\(\times\)C3:Dic3 Direct product of C2 and C3⋊Dic3 2, 3
72 [72, 35] C32:7D4 2nd semidirect product of C32 and D4 acting via D4/C22=C2 2, 3
72 [72, 36] C6\(\times\)C12 Abelian group of type [6, 12] 2, 3
72 [72, 37] D4\(\times\)C32 Direct product of C32 and D4 2, 3
72 [72, 38] Q8\(\times\)C32 Direct product of C32 and Q8 2, 3
72 [72, 39] F9 Frobenius group 2, 3
72 [72, 40] S3wrC2 Wreath product of S3 by C2 2, 3
72 [72, 41] PSU(3, 2) Projective special unitary group on 𝔽23 2, 3
72 [72, 42] C3\(\times\)S4 Direct product of C3 and S4 2, 3
72 [72, 43] C3:S4 The semidirect product of C3 and S4 acting via S4/A4=C2 2, 3
72 [72, 44] S3\(\times\)A4 Direct product of S3 and A4 2, 3
72 [72, 45] C2\(\times\)C32:C4 Direct product of C2 and C32⋊C4 2, 3
72 [72, 46] C2\(\times\)S32 Direct product of C2, S3 and S3 2, 3
72 [72, 47] C6\(\times\)A4 Direct product of C6 and A4 2, 3
72 [72, 48] S3\(\times\)C2\(\times\)C6 Direct product of C2×C6 and S3 2, 3
72 [72, 49] C22\(\times\)C3:S3 Direct product of C22 and C3⋊S3 2, 3
72 [72, 50] C2\(\times\)C62 Abelian group of type [2, 6,6] 2, 3
73 [73, 1] C73 Cyclic group 73
74 [74, 1] D37 Dihedral group 2, 37
74 [74, 2] C74 Cyclic group 2, 37
75 [75, 1] C75 Cyclic group 3, 5
75 [75, 2] C52:C3 The semidirect product of C52 and C3 acting faithfully 3, 5
75 [75, 3] C5\(\times\)C15 Abelian group of type [5, 15] 3, 5
76 [76, 1] Dic19 Dicyclic group 2, 19
76 [76, 2] C76 Cyclic group 2, 19
76 [76, 3] D38 Dihedral group 2, 19
76 [76, 4] C2\(\times\)C38 Abelian group of type [2, 38] 2, 19
77 [77, 1] C77 Cyclic group 7, 11
78 [78, 1] C13:C6 The semidirect product of C13 and C6 acting faithfully 2, 3, 13
78 [78, 2] C2\(\times\)C13:C3 Direct product of C2 and C13⋊C3 2, 3, 13
78 [78, 3] S3\(\times\)C13 Direct product of C13 and S3 2, 3, 13
78 [78, 4] C3\(\times\)D13 Direct product of C3 and D13 2, 3, 13
78 [78, 5] D39 Dihedral group 2, 3, 13
78 [78, 6] C78 Cyclic group 2, 3, 13
79 [79, 1] C79 Cyclic group 79
80 [80, 1] C5:2C16 The semidirect product of C5 and C16 acting via C16/C8=C2 2, 5
80 [80, 2] C80 Cyclic group 2, 5
80 [80, 3] C5:C16 The semidirect product of C5 and C16 acting via C16/C4=C4 2, 5
80 [80, 4] C8\(\times\)D5 Direct product of C8 and D5 2, 5
80 [80, 5] C8:D5 3rd semidirect product of C8 and D5 acting via D5/C5=C2 2, 5
80 [80, 6] C40:C2 2nd semidirect product of C40 and C2 acting faithfully 2, 5
80 [80, 7] D40 Dihedral group 2, 5
80 [80, 8] Dic20 Dicyclic group 2, 5
80 [80, 9] C2\(\times\)C5:2C8 Direct product of C2 and C5⋊2C8 2, 5
80 [80, 10] C4.Dic5 The non-split extension by C4 of Dic5 acting via Dic5/C10=C2 2, 5
80 [80, 11] C4\(\times\)Dic5 Direct product of C4 and Dic5 2, 5
80 [80, 12] C10.D4 1st non-split extension by C10 of D4 acting via D4/C22=C2 2, 5
80 [80, 13] C4:Dic5 The semidirect product of C4 and Dic5 acting via Dic5/C10=C2 2, 5
80 [80, 14] D10:C4 1st semidirect product of D10 and C4 acting via C4/C2=C2 2, 5
80 [80, 15] D4:D5 The semidirect product of D4 and D5 acting via D5/C5=C2 2, 5
80 [80, 16] D4.D5 The non-split extension by D4 of D5 acting via D5/C5=C2 2, 5
80 [80, 17] Q8:D5 The semidirect product of Q8 and D5 acting via D5/C5=C2 2, 5
80 [80, 18] C5:Q16 The semidirect product of C5 and Q16 acting via Q16/Q8=C2 2, 5
80 [80, 19] C23.D5 The non-split extension by C23 of D5 acting via D5/C5=C2 2, 5
80 [80, 20] C4\(\times\)C20 Abelian group of type [4, 20] 2, 5
80 [80, 21] C5\(\times\)C22:C4 Direct product of C5 and C22⋊C4 2, 5
80 [80, 22] C5\(\times\)C4:C4 Direct product of C5 and C4⋊C4 2, 5
80 [80, 23] C2\(\times\)C40 Abelian group of type [2, 40] 2, 5
80 [80, 24] C5\(\times\)M4(2) Direct product of C5 and M4(2) 2, 5
80 [80, 25] C5\(\times\)D8 Direct product of C5 and D8 2, 5
80 [80, 26] C5\(\times\)SD16 Direct product of C5 and SD16 2, 5
80 [80, 27] C5\(\times\)Q16 Direct product of C5 and Q16 2, 5
80 [80, 28] D5:C8 The semidirect product of D5 and C8 acting via C8/C4=C2 2, 5
80 [80, 29] C4.F5 The non-split extension by C4 of F5 acting via F5/D5=C2 2, 5
80 [80, 30] C4\(\times\)F5 Direct product of C4 and F5 2, 5
80 [80, 31] C4:F5 The semidirect product of C4 and F5 acting via F5/D5=C2 2, 5
80 [80, 32] C2\(\times\)C5:C8 Direct product of C2 and C5⋊C8 2, 5
80 [80, 33] C22.F5 The non-split extension by C22 of F5 acting via F5/D5=C2 2, 5
80 [80, 34] C22:F5 The semidirect product of C22 and F5 acting via F5/D5=C2 2, 5
80 [80, 35] C2\(\times\)Dic10 Direct product of C2 and Dic10 2, 5
80 [80, 36] C2\(\times\)C4\(\times\)D5 Direct product of C2×C4 and D5 2, 5
80 [80, 37] C2\(\times\)D20 Direct product of C2 and D20 2, 5
80 [80, 38] C4oD20 Central product of C4 and D20 2, 5
80 [80, 39] D4\(\times\)D5 Direct product of D4 and D5 2, 5
80 [80, 40] D4:2D5 The semidirect product of D4 and D5 acting through Inn(D4) 2, 5
80 [80, 41] Q8\(\times\)D5 Direct product of Q8 and D5 2, 5
80 [80, 42] Q8:2D5 The semidirect product of Q8 and D5 acting through Inn(Q8) 2, 5
80 [80, 43] C22\(\times\)Dic5 Direct product of C22 and Dic5 2, 5
80 [80, 44] C2\(\times\)C5:D4 Direct product of C2 and C5⋊D4 2, 5
80 [80, 45] C22\(\times\)C20 Abelian group of type [2, 2,20] 2, 5
80 [80, 46] D4\(\times\)C10 Direct product of C10 and D4 2, 5
80 [80, 47] Q8\(\times\)C10 Direct product of C10 and Q8 2, 5
80 [80, 48] C5\(\times\)C4oD4 Direct product of C5 and C4○D4 2, 5
80 [80, 49] C24:C5 The semidirect product of C24 and C5 acting faithfully 2, 5
80 [80, 50] C22\(\times\)F5 Direct product of C22 and F5 2, 5
80 [80, 51] C23\(\times\)D5 Direct product of C23 and D5 2, 5
80 [80, 52] C23\(\times\)C10 Abelian group of type [2, 2,2, 10] 2, 5
81 [81, 1] C81 Cyclic group 3
81 [81, 2] C92 Abelian group of type [9, 9] 3
81 [81, 3] C32:C9 The semidirect product of C32 and C9 acting via C9/C3=C3 3
81 [81, 4] C9:C9 The semidirect product of C9 and C9 acting via C9/C3=C3 3
81 [81, 5] C3\(\times\)C27 Abelian group of type [3, 27] 3
81 [81, 6] C27:C3 The semidirect product of C27 and C3 acting faithfully 3
81 [81, 7] C3wrC3 Wreath product of C3 by C3 3
81 [81, 8] He3.C3 The non-split extension by He3 of C3 acting faithfully 3
81 [81, 9] He3:C3 2nd semidirect product of He3 and C3 acting faithfully 3
81 [81, 10] C3.He3 4th central stem extension by C3 of He3 3
81 [81, 11] C32\(\times\)C9 Abelian group of type [3, 3,9] 3
81 [81, 12] C3\(\times\)He3 Direct product of C3 and He3 3
81 [81, 13] C3\(\times\)ES-(3, 1) Direct product of C3 and 3- 1+2 3
81 [81, 14] C9oHe3 Central product of C9 and He3 3
81 [81, 15] C34 Elementary abelian group of type [3, 3,3, 3] 3
82 [82, 1] D41 Dihedral group 2, 41
82 [82, 2] C82 Cyclic group 2, 41
98 [98, 4] C7:D7 Frobenius group (C7\(\times\)C7)\(\rtimes\)C2 7
100 [100, 11] (C5:D5).C2 Frobenius group (C5\(\times\)C5)\(\rtimes\)C4 5
147 [147, 4] (C7\(\times\)C7)\(\rtimes\)C3 Frobenius group (C7\(\times\)C7)\(\rtimes\)C3 7
242 [242, 4] (C11\(\times\)C11)\(\rtimes\)C2 Frobenius group (C11\(\times\)C11)\(\rtimes\)C2 11
294 [294, 13] (C7\(\times\)C7)\(\rtimes\)C6 Frobenius group (C7\(\times\)C7)\(\rtimes\)C6 7
338 [338, 4] (C13\(\times\)C13)\(\rtimes\)C2 Frobenius group (C13\(\times\)C13)\(\rtimes\)C2 13
507 [507, 4] (C13\(\times\)C13)\(\rtimes\)C3 Frobenius group (C13\(\times\)C13)\(\rtimes\)C3 13
600 [600, 149] (C7\(\times\) C7)\(\rtimes\)C24 Frobenius group (C5\(\times\)C5)\(\rtimes\)C24 5
605 [605, 4] (C11\(\times\)C11)\(\rtimes\)C5 Frobenius group (C11\(\times\)C11)\(\rtimes\)C5 11
676 [676, 10] (C13\(\times\)C13)\(\rtimes\)C4 Frobenius group (C13\(\times\)C13)\(\rtimes\)C4 13
784 [784, 160] (C7\(\times\)C7)\(\rtimes\)C16 Frobenius group (C7\(\times\)C7)\(\rtimes\)C16 7
1014 [1014, 9] (C13\(\times\)C13)\(\rtimes\)C6 Frobenius group (C13\(\times\)C13)\(\rtimes\)C6 13
1210 [1210, 9] (C11\(\times\)C11)\(\rtimes\)C10 Frobenius group (C11\(\times\)C11)\(\rtimes\)C10 11
2028 [2028, 38] (C13\(\times\)C13)\(\rtimes\)C12 Frobenius group (C13\(\times\)C13)\(\rtimes\)C12 13
VERSION 1 / HTML5-CSS-BOOTSTRAP4

Contributors: Bernhard Böhmler, Niamh Farrell, Caroline Lassueur, Jaikrishna Patil

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