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Label Name Order Exponent $\card{\mathrm{conj}(G)}$ Center Central quotient Commutator Abelianization Type - length
60.1 $C_3:C_{20}$ $2^{2} \cdot 3 \cdot 5$ $2^{2} \cdot 3 \cdot 5$ 30 $C_{10}$ $S_3$ $C_3$ $C_{20}$ Solvable - 2
60.2 $C_5:C_{12}$ $2^{2} \cdot 3 \cdot 5$ $2^{2} \cdot 3 \cdot 5$ 24 $C_6$ $D_5$ $C_5$ $C_{12}$ Solvable - 2
60.3 $C_{15}:C_4$ $2^{2} \cdot 3 \cdot 5$ $2^{2} \cdot 3 \cdot 5$ 18 $C_2$ $D_{15}$ $C_{15}$ $C_{4}$ Solvable - 2
60.6 $C_3\times F_5$ $2^{2} \cdot 3 \cdot 5$ $2^{2} \cdot 3 \cdot 5$ 15 $C_3$ $F_5$ $C_5$ $C_{12}$ Solvable - 2
60.7 $C_{15}:C_4$ $2^{2} \cdot 3 \cdot 5$ $2^{2} \cdot 3 \cdot 5$ 9 $C_1$ $C_{15}:C_4$ $C_{15}$ $C_{4}$ Solvable - 2
60.10 $C_3\times D_{10}$ $2^{2} \cdot 3 \cdot 5$ $2 \cdot 3 \cdot 5$ 24 $C_6$ $D_5$ $C_5$ $C_{2} \times C_{6}$ Solvable - 2
60.11 $S_3\times C_{10}$ $2^{2} \cdot 3 \cdot 5$ $2 \cdot 3 \cdot 5$ 30 $C_{10}$ $S_3$ $C_3$ $C_{2} \times C_{10}$ Solvable - 2
60.12 $D_{30}$ $2^{2} \cdot 3 \cdot 5$ $2 \cdot 3 \cdot 5$ 18 $C_2$ $D_{15}$ $C_{15}$ $C_{2}^{2}$ Solvable - 2
84.1 $C_7:C_{12}$ $2^{2} \cdot 3 \cdot 7$ $2^{2} \cdot 3 \cdot 7$ 14 $C_2$ $F_7$ $C_7$ $C_{12}$ Solvable - 2
84.2 $C_7:C_{12}$ $2^{2} \cdot 3 \cdot 7$ $2^{2} \cdot 3 \cdot 7$ 20 $C_4$ $C_7:C_3$ $C_7$ $C_{12}$ Solvable - 2
84.3 $C_3:C_{28}$ $2^{2} \cdot 3 \cdot 7$ $2^{2} \cdot 3 \cdot 7$ 42 $C_{14}$ $S_3$ $C_3$ $C_{28}$ Solvable - 2
84.4 $C_7:C_{12}$ $2^{2} \cdot 3 \cdot 7$ $2^{2} \cdot 3 \cdot 7$ 30 $C_6$ $D_7$ $C_7$ $C_{12}$ Solvable - 2
84.5 $C_{21}:C_4$ $2^{2} \cdot 3 \cdot 7$ $2^{2} \cdot 3 \cdot 7$ 24 $C_2$ $D_{21}$ $C_{21}$ $C_{4}$ Solvable - 2
84.7 $C_2\times F_7$ $2^{2} \cdot 3 \cdot 7$ $2 \cdot 3 \cdot 7$ 14 $C_2$ $F_7$ $C_7$ $C_{2} \times C_{6}$ Solvable - 2
84.9 $C_{14}:C_6$ $2^{2} \cdot 3 \cdot 7$ $2 \cdot 3 \cdot 7$ 20 $C_2^2$ $C_7:C_3$ $C_7$ $C_{2} \times C_{6}$ Solvable - 2
84.12 $C_3\times D_{14}$ $2^{2} \cdot 3 \cdot 7$ $2 \cdot 3 \cdot 7$ 30 $C_6$ $D_7$ $C_7$ $C_{2} \times C_{6}$ Solvable - 2
84.13 $S_3\times C_{14}$ $2^{2} \cdot 3 \cdot 7$ $2 \cdot 3 \cdot 7$ 42 $C_{14}$ $S_3$ $C_3$ $C_{2} \times C_{14}$ Solvable - 2
84.14 $D_{42}$ $2^{2} \cdot 3 \cdot 7$ $2 \cdot 3 \cdot 7$ 24 $C_2$ $D_{21}$ $C_{21}$ $C_{2}^{2}$ Solvable - 2
90.1 $C_5\times D_9$ $2 \cdot 3^{2} \cdot 5$ $2 \cdot 3^{2} \cdot 5$ 30 $C_5$ $D_9$ $C_9$ $C_{10}$ Solvable - 2
90.2 $C_9\times D_5$ $2 \cdot 3^{2} \cdot 5$ $2 \cdot 3^{2} \cdot 5$ 36 $C_9$ $D_5$ $C_5$ $C_{18}$ Solvable - 2
90.3 $D_{45}$ $2 \cdot 3^{2} \cdot 5$ $2 \cdot 3^{2} \cdot 5$ 24 $C_1$ $D_{45}$ $C_{45}$ $C_{2}$ Solvable - 2
90.5 $C_3^2\times D_5$ $2 \cdot 3^{2} \cdot 5$ $2 \cdot 3 \cdot 5$ 36 $C_3^2$ $D_5$ $C_5$ $C_{3} \times C_{6}$ Solvable - 2
90.6 $S_3\times C_{15}$ $2 \cdot 3^{2} \cdot 5$ $2 \cdot 3 \cdot 5$ 45 $C_{15}$ $S_3$ $C_3$ $C_{30}$ Solvable - 2
90.7 $C_3\times D_{15}$ $2 \cdot 3^{2} \cdot 5$ $2 \cdot 3 \cdot 5$ 27 $C_3$ $D_{15}$ $C_{15}$ $C_{6}$ Solvable - 2
126.1 $C_7:C_{18}$ $2 \cdot 3^{2} \cdot 7$ $2 \cdot 3^{2} \cdot 7$ 21 $C_3$ $F_7$ $C_7$ $C_{18}$ Solvable - 2
126.2 $C_7:C_{18}$ $2 \cdot 3^{2} \cdot 7$ $2 \cdot 3^{2} \cdot 7$ 30 $C_6$ $C_7:C_3$ $C_7$ $C_{18}$ Solvable - 2
126.3 $C_7\times D_9$ $2 \cdot 3^{2} \cdot 7$ $2 \cdot 3^{2} \cdot 7$ 42 $C_7$ $D_9$ $C_9$ $C_{14}$ Solvable - 2
126.4 $C_9\times D_7$ $2 \cdot 3^{2} \cdot 7$ $2 \cdot 3^{2} \cdot 7$ 45 $C_9$ $D_7$ $C_7$ $C_{18}$ Solvable - 2
126.5 $D_{63}$ $2 \cdot 3^{2} \cdot 7$ $2 \cdot 3^{2} \cdot 7$ 33 $C_1$ $D_{63}$ $C_{63}$ $C_{2}$ Solvable - 2
126.7 $C_3\times F_7$ $2 \cdot 3^{2} \cdot 7$ $2 \cdot 3 \cdot 7$ 21 $C_3$ $F_7$ $C_7$ $C_{3} \times C_{6}$ Solvable - 2
126.8 $C_{21}:C_6$ $2 \cdot 3^{2} \cdot 7$ $2 \cdot 3 \cdot 7$ 15 $C_1$ $C_{21}:C_6$ $C_{21}$ $C_{6}$ Solvable - 2
126.9 $C_{21}:C_6$ $2 \cdot 3^{2} \cdot 7$ $2 \cdot 3 \cdot 7$ 12 $C_1$ $C_{21}:C_6$ $C_{21}$ $C_{6}$ Solvable - 2
126.10 $C_{21}:C_6$ $2 \cdot 3^{2} \cdot 7$ $2 \cdot 3 \cdot 7$ 30 $C_6$ $C_7:C_3$ $C_7$ $C_{3} \times C_{6}$ Solvable - 2
126.11 $C_3^2\times D_7$ $2 \cdot 3^{2} \cdot 7$ $2 \cdot 3 \cdot 7$ 45 $C_3^2$ $D_7$ $C_7$ $C_{3} \times C_{6}$ Solvable - 2
126.12 $S_3\times C_{21}$ $2 \cdot 3^{2} \cdot 7$ $2 \cdot 3 \cdot 7$ 63 $C_{21}$ $S_3$ $C_3$ $C_{42}$ Solvable - 2
126.13 $C_3\times D_{21}$ $2 \cdot 3^{2} \cdot 7$ $2 \cdot 3 \cdot 7$ 36 $C_3$ $D_{21}$ $C_{21}$ $C_{6}$ Solvable - 2
132.1 $C_3:C_{44}$ $2^{2} \cdot 3 \cdot 11$ $2^{2} \cdot 3 \cdot 11$ 66 $C_{22}$ $S_3$ $C_3$ $C_{44}$ Solvable - 2
132.2 $C_{11}:C_{12}$ $2^{2} \cdot 3 \cdot 11$ $2^{2} \cdot 3 \cdot 11$ 42 $C_6$ $D_{11}$ $C_{11}$ $C_{12}$ Solvable - 2
132.3 $C_{33}:C_4$ $2^{2} \cdot 3 \cdot 11$ $2^{2} \cdot 3 \cdot 11$ 36 $C_2$ $D_{33}$ $C_{33}$ $C_{4}$ Solvable - 2
132.7 $C_3\times D_{22}$ $2^{2} \cdot 3 \cdot 11$ $2 \cdot 3 \cdot 11$ 42 $C_6$ $D_{11}$ $C_{11}$ $C_{2} \times C_{6}$ Solvable - 2
132.8 $S_3\times C_{22}$ $2^{2} \cdot 3 \cdot 11$ $2 \cdot 3 \cdot 11$ 66 $C_{22}$ $S_3$ $C_3$ $C_{2} \times C_{22}$ Solvable - 2
132.9 $D_{66}$ $2^{2} \cdot 3 \cdot 11$ $2 \cdot 3 \cdot 11$ 36 $C_2$ $D_{33}$ $C_{33}$ $C_{2}^{2}$ Solvable - 2
140.1 $C_5:C_{28}$ $2^{2} \cdot 5 \cdot 7$ $2^{2} \cdot 5 \cdot 7$ 56 $C_{14}$ $D_5$ $C_5$ $C_{28}$ Solvable - 2
140.2 $C_7:C_{20}$ $2^{2} \cdot 5 \cdot 7$ $2^{2} \cdot 5 \cdot 7$ 50 $C_{10}$ $D_7$ $C_7$ $C_{20}$ Solvable - 2
140.3 $C_{35}:C_4$ $2^{2} \cdot 5 \cdot 7$ $2^{2} \cdot 5 \cdot 7$ 38 $C_2$ $D_{35}$ $C_{35}$ $C_{4}$ Solvable - 2
140.5 $C_7\times F_5$ $2^{2} \cdot 5 \cdot 7$ $2^{2} \cdot 5 \cdot 7$ 35 $C_7$ $F_5$ $C_5$ $C_{28}$ Solvable - 2
140.6 $C_{35}:C_4$ $2^{2} \cdot 5 \cdot 7$ $2^{2} \cdot 5 \cdot 7$ 17 $C_1$ $C_{35}:C_4$ $C_{35}$ $C_{4}$ Solvable - 2
140.8 $C_5\times D_{14}$ $2^{2} \cdot 5 \cdot 7$ $2 \cdot 5 \cdot 7$ 50 $C_{10}$ $D_7$ $C_7$ $C_{2} \times C_{10}$ Solvable - 2
140.9 $C_7\times D_{10}$ $2^{2} \cdot 5 \cdot 7$ $2 \cdot 5 \cdot 7$ 56 $C_{14}$ $D_5$ $C_5$ $C_{2} \times C_{14}$ Solvable - 2
140.10 $D_{70}$ $2^{2} \cdot 5 \cdot 7$ $2 \cdot 5 \cdot 7$ 38 $C_2$ $D_{35}$ $C_{35}$ $C_{2}^{2}$ Solvable - 2
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