The results below are complete, since the LMFDB contains all groups of order at most 2000 except orders larger than 500 that are multiples of 128

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Order Exponent Nilpotency class Nilpotent
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Results (1-50 of 682 matches)

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Label Name Order Exponent $\card{\mathrm{conj}(G)}$ Center Central quotient Commutator Abelianization Type - length
65.1 $C_{65}$ $5 \cdot 13$ $5 \cdot 13$ 65 $C_{65}$ $C_1$ $C_1$ $C_{65}$ Cyclic
66.1 $S_3\times C_{11}$ $2 \cdot 3 \cdot 11$ $2 \cdot 3 \cdot 11$ 33 $C_{11}$ $S_3$ $C_3$ $C_{22}$ Solvable - 2
66.2 $C_3\times D_{11}$ $2 \cdot 3 \cdot 11$ $2 \cdot 3 \cdot 11$ 21 $C_3$ $D_{11}$ $C_{11}$ $C_{6}$ Solvable - 2
66.3 $D_{33}$ $2 \cdot 3 \cdot 11$ $2 \cdot 3 \cdot 11$ 18 $C_1$ $D_{33}$ $C_{33}$ $C_{2}$ Solvable - 2
66.4 $C_{66}$ $2 \cdot 3 \cdot 11$ $2 \cdot 3 \cdot 11$ 66 $C_{66}$ $C_1$ $C_1$ $C_{66}$ Cyclic
67.1 $C_{67}$ $67$ $67$ 67 $C_{67}$ $C_1$ $C_1$ $C_{67}$ Cyclic
68.1 $C_{17}:C_4$ $2^{2} \cdot 17$ $2^{2} \cdot 17$ 20 $C_2$ $D_{17}$ $C_{17}$ $C_{4}$ Solvable - 2
68.2 $C_{68}$ $2^{2} \cdot 17$ $2^{2} \cdot 17$ 68 $C_{68}$ $C_1$ $C_1$ $C_{68}$ Cyclic
68.3 $C_{17}:C_4$ $2^{2} \cdot 17$ $2^{2} \cdot 17$ 8 $C_1$ $C_{17}:C_4$ $C_{17}$ $C_{4}$ Solvable - 2
68.4 $D_{34}$ $2^{2} \cdot 17$ $2 \cdot 17$ 20 $C_2$ $D_{17}$ $C_{17}$ $C_{2}^{2}$ Solvable - 2
68.5 $C_2\times C_{34}$ $2^{2} \cdot 17$ $2 \cdot 17$ 68 $C_2\times C_{34}$ $C_1$ $C_1$ $C_{2} \times C_{34}$ Abelian - 2
69.1 $C_{69}$ $3 \cdot 23$ $3 \cdot 23$ 69 $C_{69}$ $C_1$ $C_1$ $C_{69}$ Cyclic
70.1 $C_7\times D_5$ $2 \cdot 5 \cdot 7$ $2 \cdot 5 \cdot 7$ 28 $C_7$ $D_5$ $C_5$ $C_{14}$ Solvable - 2
70.2 $C_5\times D_7$ $2 \cdot 5 \cdot 7$ $2 \cdot 5 \cdot 7$ 25 $C_5$ $D_7$ $C_7$ $C_{10}$ Solvable - 2
70.3 $D_{35}$ $2 \cdot 5 \cdot 7$ $2 \cdot 5 \cdot 7$ 19 $C_1$ $D_{35}$ $C_{35}$ $C_{2}$ Solvable - 2
70.4 $C_{70}$ $2 \cdot 5 \cdot 7$ $2 \cdot 5 \cdot 7$ 70 $C_{70}$ $C_1$ $C_1$ $C_{70}$ Cyclic
71.1 $C_{71}$ $71$ $71$ 71 $C_{71}$ $C_1$ $C_1$ $C_{71}$ Cyclic
72.1 $C_9:C_8$ $2^{3} \cdot 3^{2}$ $2^{3} \cdot 3^{2}$ 24 $C_4$ $D_9$ $C_9$ $C_{8}$ Solvable - 2
72.2 $C_{72}$ $2^{3} \cdot 3^{2}$ $2^{3} \cdot 3^{2}$ 72 $C_{72}$ $C_1$ $C_1$ $C_{72}$ Cyclic
72.3 $Q_8:C_9$ $2^{3} \cdot 3^{2}$ $2^{2} \cdot 3^{2}$ 21 $C_6$ $A_4$ $Q_8$ $C_{9}$ Solvable - 3
72.4 $C_9:Q_8$ $2^{3} \cdot 3^{2}$ $2^{2} \cdot 3^{2}$ 21 $C_2$ $D_{18}$ $C_{18}$ $C_{2}^{2}$ Solvable - 2
72.5 $C_4\times D_9$ $2^{3} \cdot 3^{2}$ $2^{2} \cdot 3^{2}$ 24 $C_4$ $D_9$ $C_9$ $C_{2} \times C_{4}$ Solvable - 2
72.6 $D_{36}$ $2^{3} \cdot 3^{2}$ $2^{2} \cdot 3^{2}$ 21 $C_2$ $D_{18}$ $C_{18}$ $C_{2}^{2}$ Solvable - 2
72.7 $C_{18}:C_4$ $2^{3} \cdot 3^{2}$ $2^{2} \cdot 3^{2}$ 24 $C_2^2$ $D_9$ $C_9$ $C_{2} \times C_{4}$ Solvable - 2
72.8 $C_9:D_4$ $2^{3} \cdot 3^{2}$ $2^{2} \cdot 3^{2}$ 21 $C_2$ $D_{18}$ $C_{18}$ $C_{2}^{2}$ Solvable - 2
72.9 $C_2\times C_{36}$ $2^{3} \cdot 3^{2}$ $2^{2} \cdot 3^{2}$ 72 $C_2\times C_{36}$ $C_1$ $C_1$ $C_{2} \times C_{36}$ Abelian - 2
72.10 $D_4\times C_9$ $2^{3} \cdot 3^{2}$ $2^{2} \cdot 3^{2}$ 45 $C_{18}$ $C_2^2$ $C_2$ $C_{2} \times C_{18}$ Nilpotent - 2
72.11 $Q_8\times C_9$ $2^{3} \cdot 3^{2}$ $2^{2} \cdot 3^{2}$ 45 $C_{18}$ $C_2^2$ $C_2$ $C_{2} \times C_{18}$ Nilpotent - 2
72.12 $C_3:C_{24}$ $2^{3} \cdot 3^{2}$ $2^{3} \cdot 3$ 36 $C_{12}$ $S_3$ $C_3$ $C_{24}$ Solvable - 2
72.13 $C_3^2:C_8$ $2^{3} \cdot 3^{2}$ $2^{3} \cdot 3$ 24 $C_4$ $C_3:S_3$ $C_3^2$ $C_{8}$ Solvable - 2
72.14 $C_3\times C_{24}$ $2^{3} \cdot 3^{2}$ $2^{3} \cdot 3$ 72 $C_3\times C_{24}$ $C_1$ $C_1$ $C_{3} \times C_{24}$ Abelian - 2
72.15 $C_2^2:D_9$ $2^{3} \cdot 3^{2}$ $2^{2} \cdot 3^{2}$ 9 $C_1$ $C_2^2:D_9$ $C_2^2:C_9$ $C_{2}$ Solvable - 3
72.16 $C_2^2:C_{18}$ $2^{3} \cdot 3^{2}$ $2 \cdot 3^{2}$ 24 $C_6$ $A_4$ $C_2^2$ $C_{18}$ Solvable - 2
72.17 $C_2\times D_{18}$ $2^{3} \cdot 3^{2}$ $2 \cdot 3^{2}$ 24 $C_2^2$ $D_9$ $C_9$ $C_{2}^{3}$ Solvable - 2
72.18 $C_2^2\times C_{18}$ $2^{3} \cdot 3^{2}$ $2 \cdot 3^{2}$ 72 $C_2^2\times C_{18}$ $C_1$ $C_1$ $C_{2}^{2} \times C_{18}$ Abelian - 3
72.19 $C_3^2:C_8$ $2^{3} \cdot 3^{2}$ $2^{3} \cdot 3$ 12 $C_2$ $C_3^2:C_4$ $C_3^2$ $C_{8}$ Solvable - 2
72.20 $C_6.D_6$ $2^{3} \cdot 3^{2}$ $2^{2} \cdot 3$ 18 $C_2$ $S_3^2$ $C_3^2$ $C_{2} \times C_{4}$ Solvable - 2
72.21 $C_6.D_6$ $2^{3} \cdot 3^{2}$ $2^{2} \cdot 3$ 18 $C_2$ $S_3^2$ $C_3^2$ $C_{2} \times C_{4}$ Solvable - 2
72.22 $D_6:S_3$ $2^{3} \cdot 3^{2}$ $2^{2} \cdot 3$ 15 $C_2$ $S_3^2$ $C_3\times C_6$ $C_{2}^{2}$ Solvable - 2
72.23 $C_3:D_{12}$ $2^{3} \cdot 3^{2}$ $2^{2} \cdot 3$ 15 $C_2$ $S_3^2$ $C_3\times C_6$ $C_{2}^{2}$ Solvable - 2
72.24 $C_3^2:Q_8$ $2^{3} \cdot 3^{2}$ $2^{2} \cdot 3$ 15 $C_2$ $S_3^2$ $C_3\times C_6$ $C_{2}^{2}$ Solvable - 2
72.25 $C_3\times \SL(2,3)$ $2^{3} \cdot 3^{2}$ $2^{2} \cdot 3$ 21 $C_6$ $A_4$ $Q_8$ $C_{3}^{2}$ Solvable - 3
72.26 $C_3^2:Q_8$ $2^{3} \cdot 3^{2}$ $2^{2} \cdot 3$ 27 $C_6$ $D_6$ $C_6$ $C_{2} \times C_{6}$ Solvable - 2
72.27 $S_3\times C_{12}$ $2^{3} \cdot 3^{2}$ $2^{2} \cdot 3$ 36 $C_{12}$ $S_3$ $C_3$ $C_{2} \times C_{12}$ Solvable - 2
72.28 $C_3\times D_{12}$ $2^{3} \cdot 3^{2}$ $2^{2} \cdot 3$ 27 $C_6$ $D_6$ $C_6$ $C_{2} \times C_{6}$ Solvable - 2
72.29 $C_6:C_{12}$ $2^{3} \cdot 3^{2}$ $2^{2} \cdot 3$ 36 $C_2\times C_6$ $S_3$ $C_3$ $C_{2} \times C_{12}$ Solvable - 2
72.30 $C_6\wr C_2$ $2^{3} \cdot 3^{2}$ $2^{2} \cdot 3$ 27 $C_6$ $D_6$ $C_6$ $C_{2} \times C_{6}$ Solvable - 2
72.31 $C_3^2:Q_8$ $2^{3} \cdot 3^{2}$ $2^{2} \cdot 3$ 21 $C_2$ $C_6:S_3$ $C_3\times C_6$ $C_{2}^{2}$ Solvable - 2
72.32 $C_{12}:S_3$ $2^{3} \cdot 3^{2}$ $2^{2} \cdot 3$ 24 $C_4$ $C_3:S_3$ $C_3^2$ $C_{2} \times C_{4}$ Solvable - 2
72.33 $C_3:D_{12}$ $2^{3} \cdot 3^{2}$ $2^{2} \cdot 3$ 21 $C_2$ $C_6:S_3$ $C_3\times C_6$ $C_{2}^{2}$ Solvable - 2
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