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Label Name Family name Order Exponent $\card{\mathrm{conj}(G)}$ Center Central quotient Commutator Abelianization Type - length
2.1 $C_2$ $F_{2}$ $2$ $2$ 2 $C_2$ $C_1$ $C_1$ $C_{2}$ Cyclic
6.1 $S_3$ $F_{3}$ $2 \cdot 3$ $2 \cdot 3$ 3 $C_1$ $S_3$ $C_3$ $C_{2}$ Solvable - 2
12.3 $A_4$ $F_{4}$ $2^{2} \cdot 3$ $2 \cdot 3$ 4 $C_1$ $A_4$ $C_2^2$ $C_{3}$ Solvable - 2
20.3 $F_5$ $F_{5}$ $2^{2} \cdot 5$ $2^{2} \cdot 5$ 5 $C_1$ $F_5$ $C_5$ $C_{4}$ Solvable - 2
42.1 $F_7$ $F_{7}$ $2 \cdot 3 \cdot 7$ $2 \cdot 3 \cdot 7$ 7 $C_1$ $F_7$ $C_7$ $C_{6}$ Solvable - 2
56.11 $F_8$ $F_{8}$ $2^{3} \cdot 7$ $2 \cdot 7$ 8 $C_1$ $F_8$ $C_2^3$ $C_{7}$ Solvable - 2
72.39 $F_9$ $F_{9}$ $2^{3} \cdot 3^{2}$ $2^{3} \cdot 3$ 9 $C_1$ $F_9$ $C_3^2$ $C_{8}$ Solvable - 2
110.1 $F_{11}$ $F_{11}$ $2 \cdot 5 \cdot 11$ $2 \cdot 5 \cdot 11$ 11 $C_1$ $F_{11}$ $C_{11}$ $C_{10}$ Solvable - 2
156.7 $F_{13}$ $F_{13}$ $2^{2} \cdot 3 \cdot 13$ $2^{2} \cdot 3 \cdot 13$ 13 $C_1$ $F_{13}$ $C_{13}$ $C_{12}$ Solvable - 2
240.191 $F_{16}$ $F_{16}$ $2^{4} \cdot 3 \cdot 5$ $2 \cdot 3 \cdot 5$ 16 $C_1$ $F_{16}$ $C_2^4$ $C_{15}$ Solvable - 2
272.50 $F_{17}$ $F_{17}$ $2^{4} \cdot 17$ $2^{4} \cdot 17$ 17 $C_1$ $F_{17}$ $C_{17}$ $C_{16}$ Solvable - 2
342.7 $F_{19}$ $F_{19}$ $2 \cdot 3^{2} \cdot 19$ $2 \cdot 3^{2} \cdot 19$ 19 $C_1$ $F_{19}$ $C_{19}$ $C_{18}$ Solvable - 2
506.1 $F_{23}$ $F_{23}$ $2 \cdot 11 \cdot 23$ $2 \cdot 11 \cdot 23$ 23 $C_1$ $F_{23}$ $C_{23}$ $C_{22}$ Solvable - 2
600.149 $F_{25}$ $F_{25}$ $2^{3} \cdot 3 \cdot 5^{2}$ $2^{3} \cdot 3 \cdot 5$ 25 $C_1$ $F_{25}$ $C_5^2$ $C_{24}$ Solvable - 2
702.47 $F_{27}$ $F_{27}$ $2 \cdot 3^{3} \cdot 13$ $2 \cdot 3 \cdot 13$ 27 $C_1$ $F_{27}$ $C_3^3$ $C_{26}$ Solvable - 2
812.7 $F_{29}$ $F_{29}$ $2^{2} \cdot 7 \cdot 29$ $2^{2} \cdot 7 \cdot 29$ 29 $C_1$ $F_{29}$ $C_{29}$ $C_{28}$ Solvable - 2
930.1 $F_{31}$ $F_{31}$ $2 \cdot 3 \cdot 5 \cdot 31$ $2 \cdot 3 \cdot 5 \cdot 31$ 31 $C_1$ $F_{31}$ $C_{31}$ $C_{30}$ Solvable - 2
992.194 $F_{32}$ $F_{32}$ $2^{5} \cdot 31$ $2 \cdot 31$ 32 $C_1$ $F_{32}$ $C_2^5$ $C_{31}$ Solvable - 2
1332.31 $F_{37}$ $F_{37}$ $2^{2} \cdot 3^{2} \cdot 37$ $2^{2} \cdot 3^{2} \cdot 37$ 37 $C_1$ $F_{37}$ $C_{37}$ $C_{36}$ Solvable - 2
1640.47 $F_{41}$ $F_{41}$ $2^{3} \cdot 5 \cdot 41$ $2^{3} \cdot 5 \cdot 41$ 41 $C_1$ $F_{41}$ $C_{41}$ $C_{40}$ Solvable - 2
1806.1 $F_{43}$ $F_{43}$ $2 \cdot 3 \cdot 7 \cdot 43$ $2 \cdot 3 \cdot 7 \cdot 43$ 43 $C_1$ $F_{43}$ $C_{43}$ $C_{42}$ Solvable - 2
2352.b $F_{49}$ $F_{49}$ $2^{4} \cdot 3 \cdot 7^{2}$ $2^{4} \cdot 3 \cdot 7$ 49 $C_1$ $F_{49}$ $C_7^2$ $C_{48}$ Solvable - 2
3660.a $F_{61}$ $F_{61}$ $2^{2} \cdot 3 \cdot 5 \cdot 61$ $2^{2} \cdot 3 \cdot 5 \cdot 61$ 61 $C_1$ $F_{61}$ $C_{61}$ $C_{60}$ Solvable - 2
4032.n $F_{64}$ $F_{64}$ $2^{6} \cdot 3^{2} \cdot 7$ $2 \cdot 3^{2} \cdot 7$ 64 $C_1$ $F_{64}$ $C_2^6$ $C_{63}$ Solvable - 2
5256.b $F_{73}$ $F_{73}$ $2^{3} \cdot 3^{2} \cdot 73$ $2^{3} \cdot 3^{2} \cdot 73$ 73 $C_1$ $F_{73}$ $C_{73}$ $C_{72}$ Solvable - 2
6480.a $F_{81}$ $F_{81}$ $2^{4} \cdot 3^{4} \cdot 5$ $2^{4} \cdot 3 \cdot 5$ 81 $C_1$ $F_{81}$ $C_3^4$ $C_{80}$ Solvable - 2
7832.b $F_{89}$ $F_{89}$ $2^{3} \cdot 11 \cdot 89$ $2^{3} \cdot 11 \cdot 89$ 89 $C_1$ $F_{89}$ $C_{89}$ $C_{88}$ Solvable - 2
9312.a $F_{97}$ $F_{97}$ $2^{5} \cdot 3 \cdot 97$ $2^{5} \cdot 3 \cdot 97$ 97 $C_1$ $F_{97}$ $C_{97}$ $C_{96}$ Solvable - 2
10100.a $F_{101}$ $F_{101}$ $2^{2} \cdot 5^{2} \cdot 101$ $2^{2} \cdot 5^{2} \cdot 101$ 101 $C_1$ $F_{101}$ $C_{101}$ $C_{100}$ Solvable - 2
11772.b $F_{109}$ $F_{109}$ $2^{2} \cdot 3^{3} \cdot 109$ $2^{2} \cdot 3^{3} \cdot 109$ 109 $C_1$ $F_{109}$ $C_{109}$ $C_{108}$ Solvable - 2
12656.c $F_{113}$ $F_{113}$ $2^{4} \cdot 7 \cdot 113$ $2^{4} \cdot 7 \cdot 113$ 113 $C_1$ $F_{113}$ $C_{113}$ $C_{112}$ Solvable - 2
14520.b $F_{121}$ $F_{121}$ $2^{3} \cdot 3 \cdot 5 \cdot 11^{2}$ $2^{3} \cdot 3 \cdot 5 \cdot 11$ 121 $C_1$ $F_{121}$ $C_{11}^2$ $C_{120}$ Solvable - 2
15500.d $F_{125}$ $F_{125}$ $2^{2} \cdot 5^{3} \cdot 31$ $2^{2} \cdot 5 \cdot 31$ 125 $C_1$ $F_{125}$ $C_5^3$ $C_{124}$ Solvable - 2
16002.c $F_{127}$ $F_{127}$ $2 \cdot 3^{2} \cdot 7 \cdot 127$ $2 \cdot 3^{2} \cdot 7 \cdot 127$ 127 $C_1$ $F_{127}$ $C_{127}$ $C_{126}$ Solvable - 2
16256.19325 $F_{128}$ $F_{128}$ $2^{7} \cdot 127$ $2 \cdot 127$ 128 $C_1$ $F_{128}$ $C_2^7$ $C_{127}$ Solvable - 2
18632.c $F_{137}$ $F_{137}$ $2^{3} \cdot 17 \cdot 137$ $2^{3} \cdot 17 \cdot 137$ 137 $C_1$ $F_{137}$ $C_{137}$ $C_{136}$ Solvable - 2
22650.b $F_{151}$ $F_{151}$ $2 \cdot 3 \cdot 5^{2} \cdot 151$ $2 \cdot 3 \cdot 5^{2} \cdot 151$ 151 $C_1$ $F_{151}$ $C_{151}$ $C_{150}$ Solvable - 2
24492.a $F_{157}$ $F_{157}$ $2^{2} \cdot 3 \cdot 13 \cdot 157$ $2^{2} \cdot 3 \cdot 13 \cdot 157$ 157 $C_1$ $F_{157}$ $C_{157}$ $C_{156}$ Solvable - 2
26406.a $F_{163}$ $F_{163}$ $2 \cdot 3^{4} \cdot 163$ $2 \cdot 3^{4} \cdot 163$ 163 $C_1$ $F_{163}$ $C_{163}$ $C_{162}$ Solvable - 2
28392.a $F_{169}$ $F_{169}$ $2^{3} \cdot 3 \cdot 7 \cdot 13^{2}$ $2^{3} \cdot 3 \cdot 7 \cdot 13$ 169 $C_1$ $F_{169}$ $C_{13}^2$ $C_{168}$ Solvable - 2
32580.a $F_{181}$ $F_{181}$ $2^{2} \cdot 3^{2} \cdot 5 \cdot 181$ $2^{2} \cdot 3^{2} \cdot 5 \cdot 181$ 181 $C_1$ $F_{181}$ $C_{181}$ $C_{180}$ Solvable - 2
37056.a $F_{193}$ $F_{193}$ $2^{6} \cdot 3 \cdot 193$ $2^{6} \cdot 3 \cdot 193$ 193 $C_1$ $F_{193}$ $C_{193}$ $C_{192}$ Solvable - 2
38612.a $F_{197}$ $F_{197}$ $2^{2} \cdot 7^{2} \cdot 197$ $2^{2} \cdot 7^{2} \cdot 197$ 197 $C_1$ $F_{197}$ $C_{197}$ $C_{196}$ Solvable - 2
39402.d $F_{199}$ $F_{199}$ $2 \cdot 3^{2} \cdot 11 \cdot 199$ $2 \cdot 3^{2} \cdot 11 \cdot 199$ 199 $C_1$ $F_{199}$ $C_{199}$ $C_{198}$ Solvable - 2
44310.90 $F_{211}$ $F_{211}$ $2 \cdot 3 \cdot 5 \cdot 7 \cdot 211$ $2 \cdot 3 \cdot 5 \cdot 7 \cdot 211$ 211 $C_1$ $F_{211}$ $C_{211}$ $C_{210}$ Solvable - 2
51302.1 $F_{227}$ $F_{227}$ $2 \cdot 113 \cdot 227$ $2 \cdot 113 \cdot 227$ 227 $C_1$ $F_{227}$ $C_{227}$ $C_{226}$ Solvable - 2
52212.c $F_{229}$ $F_{229}$ $2^{2} \cdot 3 \cdot 19 \cdot 229$ $2^{2} \cdot 3 \cdot 19 \cdot 229$ 229 $C_1$ $F_{229}$ $C_{229}$ $C_{228}$ Solvable - 2
54056.d $F_{233}$ $F_{233}$ $2^{3} \cdot 29 \cdot 233$ $2^{3} \cdot 29 \cdot 233$ 233 $C_1$ $F_{233}$ $C_{233}$ $C_{232}$ Solvable - 2
56882.18 $F_{239}$ $F_{239}$ $2 \cdot 7 \cdot 17 \cdot 239$ $2 \cdot 7 \cdot 17 \cdot 239$ 239 $C_1$ $F_{239}$ $C_{239}$ $C_{238}$ Solvable - 2
57840.a $F_{241}$ $F_{241}$ $2^{4} \cdot 3 \cdot 5 \cdot 241$ $2^{4} \cdot 3 \cdot 5 \cdot 241$ 241 $C_1$ $F_{241}$ $C_{241}$ $C_{240}$ Solvable - 2
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