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Label | Name | Order | Exponent | $\card{\mathrm{conj}(G)}$ | Center | Type - length |
---|---|---|---|---|---|---|
6.1 | $S_3$ | $2 \cdot 3$ | $2 \cdot 3$ | $3$ | $C_1$ | Solvable - 2 |
8.3 | $D_4$ | $2^{3}$ | $2^{2}$ | $5$ | $C_2$ | Nilpotent - 2 |
8.4 | $Q_8$ | $2^{3}$ | $2^{2}$ | $5$ | $C_2$ | Nilpotent - 2 |
10.1 | $D_5$ | $2 \cdot 5$ | $2 \cdot 5$ | $4$ | $C_1$ | Solvable - 2 |
12.1 | $C_3:C_4$ | $2^{2} \cdot 3$ | $2^{2} \cdot 3$ | $6$ | $C_2$ | Solvable - 2 |
12.3 | $A_4$ | $2^{2} \cdot 3$ | $2 \cdot 3$ | $4$ | $C_1$ | Solvable - 2 |
12.4 | $D_6$ | $2^{2} \cdot 3$ | $2 \cdot 3$ | $6$ | $C_2$ | Solvable - 2 |
14.1 | $D_7$ | $2 \cdot 7$ | $2 \cdot 7$ | $5$ | $C_1$ | Solvable - 2 |
16.3 | $C_2^2:C_4$ | $2^{4}$ | $2^{2}$ | $10$ | $C_2^2$ | Nilpotent - 2 |
16.4 | $C_4:C_4$ | $2^{4}$ | $2^{2}$ | $10$ | $C_2^2$ | Nilpotent - 2 |
16.6 | $\OD_{16}$ | $2^{4}$ | $2^{3}$ | $10$ | $C_4$ | Nilpotent - 2 |
16.7 | $D_8$ | $2^{4}$ | $2^{3}$ | $7$ | $C_2$ | Nilpotent - 3 |
16.8 | $\SD_{16}$ | $2^{4}$ | $2^{3}$ | $7$ | $C_2$ | Nilpotent - 3 |
16.9 | $Q_{16}$ | $2^{4}$ | $2^{3}$ | $7$ | $C_2$ | Nilpotent - 3 |
16.11 | $C_2\times D_4$ | $2^{4}$ | $2^{2}$ | $10$ | $C_2^2$ | Nilpotent - 2 |
16.12 | $C_2\times Q_8$ | $2^{4}$ | $2^{2}$ | $10$ | $C_2^2$ | Nilpotent - 2 |
16.13 | $D_4:C_2$ | $2^{4}$ | $2^{2}$ | $10$ | $C_4$ | Nilpotent - 2 |
18.1 | $D_9$ | $2 \cdot 3^{2}$ | $2 \cdot 3^{2}$ | $6$ | $C_1$ | Solvable - 2 |
18.3 | $C_3\times S_3$ | $2 \cdot 3^{2}$ | $2 \cdot 3$ | $9$ | $C_3$ | Solvable - 2 |
18.4 | $C_3:S_3$ | $2 \cdot 3^{2}$ | $2 \cdot 3$ | $6$ | $C_1$ | Solvable - 2 |
20.1 | $C_5:C_4$ | $2^{2} \cdot 5$ | $2^{2} \cdot 5$ | $8$ | $C_2$ | Solvable - 2 |
20.3 | $F_5$ | $2^{2} \cdot 5$ | $2^{2} \cdot 5$ | $5$ | $C_1$ | Solvable - 2 |
20.4 | $D_{10}$ | $2^{2} \cdot 5$ | $2 \cdot 5$ | $8$ | $C_2$ | Solvable - 2 |
21.1 | $C_7:C_3$ | $3 \cdot 7$ | $3 \cdot 7$ | $5$ | $C_1$ | Solvable - 2 |
22.1 | $D_{11}$ | $2 \cdot 11$ | $2 \cdot 11$ | $7$ | $C_1$ | Solvable - 2 |
24.1 | $C_3:C_8$ | $2^{3} \cdot 3$ | $2^{3} \cdot 3$ | $12$ | $C_4$ | Solvable - 2 |
24.3 | $\SL(2,3)$ | $2^{3} \cdot 3$ | $2^{2} \cdot 3$ | $7$ | $C_2$ | Solvable - 3 |
24.4 | $C_3:Q_8$ | $2^{3} \cdot 3$ | $2^{2} \cdot 3$ | $9$ | $C_2$ | Solvable - 2 |
24.5 | $C_4\times S_3$ | $2^{3} \cdot 3$ | $2^{2} \cdot 3$ | $12$ | $C_4$ | Solvable - 2 |
24.6 | $D_{12}$ | $2^{3} \cdot 3$ | $2^{2} \cdot 3$ | $9$ | $C_2$ | Solvable - 2 |
24.7 | $C_6:C_4$ | $2^{3} \cdot 3$ | $2^{2} \cdot 3$ | $12$ | $C_2^2$ | Solvable - 2 |
24.8 | $C_3:D_4$ | $2^{3} \cdot 3$ | $2^{2} \cdot 3$ | $9$ | $C_2$ | Solvable - 2 |
24.10 | $C_3\times D_4$ | $2^{3} \cdot 3$ | $2^{2} \cdot 3$ | $15$ | $C_6$ | Nilpotent - 2 |
24.11 | $C_3\times Q_8$ | $2^{3} \cdot 3$ | $2^{2} \cdot 3$ | $15$ | $C_6$ | Nilpotent - 2 |
24.12 | $S_4$ | $2^{3} \cdot 3$ | $2^{2} \cdot 3$ | $5$ | $C_1$ | Solvable - 3 |
24.13 | $C_2\times A_4$ | $2^{3} \cdot 3$ | $2 \cdot 3$ | $8$ | $C_2$ | Solvable - 2 |
24.14 | $C_2\times D_6$ | $2^{3} \cdot 3$ | $2 \cdot 3$ | $12$ | $C_2^2$ | Solvable - 2 |
26.1 | $D_{13}$ | $2 \cdot 13$ | $2 \cdot 13$ | $8$ | $C_1$ | Solvable - 2 |
27.3 | $\He_3$ | $3^{3}$ | $3$ | $11$ | $C_3$ | Nilpotent - 2 |
27.4 | $C_9:C_3$ | $3^{3}$ | $3^{2}$ | $11$ | $C_3$ | Nilpotent - 2 |
28.1 | $C_7:C_4$ | $2^{2} \cdot 7$ | $2^{2} \cdot 7$ | $10$ | $C_2$ | Solvable - 2 |
28.3 | $D_{14}$ | $2^{2} \cdot 7$ | $2 \cdot 7$ | $10$ | $C_2$ | Solvable - 2 |
30.1 | $C_5\times S_3$ | $2 \cdot 3 \cdot 5$ | $2 \cdot 3 \cdot 5$ | $15$ | $C_5$ | Solvable - 2 |
30.2 | $C_3\times D_5$ | $2 \cdot 3 \cdot 5$ | $2 \cdot 3 \cdot 5$ | $12$ | $C_3$ | Solvable - 2 |
30.3 | $D_{15}$ | $2 \cdot 3 \cdot 5$ | $2 \cdot 3 \cdot 5$ | $9$ | $C_1$ | Solvable - 2 |
32.2 | $C_2.C_4^2$ | $2^{5}$ | $2^{2}$ | $20$ | $C_2^3$ | Nilpotent - 2 |
32.4 | $C_8:C_4$ | $2^{5}$ | $2^{3}$ | $20$ | $C_2\times C_4$ | Nilpotent - 2 |
32.5 | $C_2^2:C_8$ | $2^{5}$ | $2^{3}$ | $20$ | $C_2\times C_4$ | Nilpotent - 2 |
32.6 | $C_2^3:C_4$ | $2^{5}$ | $2^{2}$ | $11$ | $C_2$ | Nilpotent - 3 |
32.7 | $\OD_{16}:C_2$ | $2^{5}$ | $2^{3}$ | $11$ | $C_2$ | Nilpotent - 3 |