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Order Exponent Nilpotency class Nilpotent
Center Commutator Central quotient Abelianization
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Perfect Direct product Semidirect product Wreath product
Automorphism group Automorphisms Min. transitive degree Min. permutation degree
Number of subgroups Num. of normal subs Num. of conj. classes
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Min. degree of $\C$-irrep Min. degree of $\Q$-irrep Min. degree of $\C$-rep Min. degree of $\Q$-rep
Outer aut. group Outer automorphisms Metabelian Metacyclic
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Results (1-50 of 236938 matches)

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Label Name Order Exponent $\card{\mathrm{conj}(G)}$ Center Central quotient Commutator Abelianization Type - length
8.5 $C_2^3$ $2^{3}$ $2$ 8 $C_2^3$ $C_1$ $C_1$ $C_{2}^{3}$ Abelian - 3
16.3 $C_2^2:C_4$ $2^{4}$ $2^{2}$ 10 $C_2^2$ $C_2^2$ $C_2$ $C_{2} \times C_{4}$ Nilpotent - 2
16.10 $C_2^2\times C_4$ $2^{4}$ $2^{2}$ 16 $C_2^2\times C_4$ $C_1$ $C_1$ $C_{2}^{2} \times C_{4}$ Abelian - 3
16.11 $C_2\times D_4$ $2^{4}$ $2^{2}$ 10 $C_2^2$ $C_2^2$ $C_2$ $C_{2}^{3}$ Nilpotent - 2
16.12 $C_2\times Q_8$ $2^{4}$ $2^{2}$ 10 $C_2^2$ $C_2^2$ $C_2$ $C_{2}^{3}$ Nilpotent - 2
16.13 $D_4:C_2$ $2^{4}$ $2^{2}$ 10 $C_4$ $C_2^2$ $C_2$ $C_{2}^{3}$ Nilpotent - 2
16.14 $C_2^4$ $2^{4}$ $2$ 16 $C_2^4$ $C_1$ $C_1$ $C_{2}^{4}$ Abelian - 4
18.4 $C_3:S_3$ $2 \cdot 3^{2}$ $2 \cdot 3$ 6 $C_1$ $C_3:S_3$ $C_3^2$ $C_{2}$ Solvable - 2
24.8 $C_3:D_4$ $2^{3} \cdot 3$ $2^{2} \cdot 3$ 9 $C_2$ $D_6$ $C_6$ $C_{2}^{2}$ Solvable - 2
24.14 $C_2\times D_6$ $2^{3} \cdot 3$ $2 \cdot 3$ 12 $C_2^2$ $S_3$ $C_3$ $C_{2}^{3}$ Solvable - 2
24.15 $C_2^2\times C_6$ $2^{3} \cdot 3$ $2 \cdot 3$ 24 $C_2^2\times C_6$ $C_1$ $C_1$ $C_{2}^{2} \times C_{6}$ Abelian - 3
27.3 $\He_3$ $3^{3}$ $3$ 11 $C_3$ $C_3^2$ $C_3$ $C_{3}^{2}$ Nilpotent - 2
27.5 $C_3^3$ $3^{3}$ $3$ 27 $C_3^3$ $C_1$ $C_1$ $C_{3}^{3}$ Abelian - 3
32.2 $C_2.C_4^2$ $2^{5}$ $2^{2}$ 20 $C_2^3$ $C_2^2$ $C_2$ $C_{4}^{2}$ Nilpotent - 2
32.5 $C_2^2:C_8$ $2^{5}$ $2^{3}$ 20 $C_2\times C_4$ $C_2^2$ $C_2$ $C_{2} \times C_{8}$ Nilpotent - 2
32.6 $C_2^3:C_4$ $2^{5}$ $2^{2}$ 11 $C_2$ $C_2^2:C_4$ $C_2^2$ $C_{2} \times C_{4}$ Nilpotent - 3
32.7 $\OD_{16}:C_2$ $2^{5}$ $2^{3}$ 11 $C_2$ $C_2^2:C_4$ $C_2^2$ $C_{2} \times C_{4}$ Nilpotent - 3
32.8 $C_4.D_4$ $2^{5}$ $2^{3}$ 11 $C_2$ $C_2^2:C_4$ $C_2^2$ $C_{2} \times C_{4}$ Nilpotent - 3
32.9 $D_4:C_4$ $2^{5}$ $2^{3}$ 14 $C_2^2$ $D_4$ $C_4$ $C_{2} \times C_{4}$ Nilpotent - 3
32.10 $Q_8:C_4$ $2^{5}$ $2^{3}$ 14 $C_2^2$ $D_4$ $C_4$ $C_{2} \times C_{4}$ Nilpotent - 3
32.11 $C_4\wr C_2$ $2^{5}$ $2^{3}$ 14 $C_4$ $D_4$ $C_4$ $C_{2} \times C_{4}$ Nilpotent - 3
32.21 $C_2\times C_4^2$ $2^{5}$ $2^{2}$ 32 $C_2\times C_4^2$ $C_1$ $C_1$ $C_{2} \times C_{4}^{2}$ Abelian - 3
32.22 $C_2^3:C_4$ $2^{5}$ $2^{2}$ 20 $C_2^3$ $C_2^2$ $C_2$ $C_{2}^{2} \times C_{4}$ Nilpotent - 2
32.23 $C_2^2.D_4$ $2^{5}$ $2^{2}$ 20 $C_2^3$ $C_2^2$ $C_2$ $C_{2}^{2} \times C_{4}$ Nilpotent - 2
32.24 $C_4^2:C_2$ $2^{5}$ $2^{2}$ 20 $C_2\times C_4$ $C_2^2$ $C_2$ $C_{2}^{2} \times C_{4}$ Nilpotent - 2
32.25 $C_4\times D_4$ $2^{5}$ $2^{2}$ 20 $C_2\times C_4$ $C_2^2$ $C_2$ $C_{2}^{2} \times C_{4}$ Nilpotent - 2
32.26 $C_4\times Q_8$ $2^{5}$ $2^{2}$ 20 $C_2\times C_4$ $C_2^2$ $C_2$ $C_{2}^{2} \times C_{4}$ Nilpotent - 2
32.27 $C_2^2\wr C_2$ $2^{5}$ $2^{2}$ 14 $C_2^2$ $C_2^3$ $C_2^2$ $C_{2}^{3}$ Nilpotent - 2
32.28 $C_4:D_4$ $2^{5}$ $2^{2}$ 14 $C_2^2$ $C_2^3$ $C_2^2$ $C_{2}^{3}$ Nilpotent - 2
32.29 $C_2^2:Q_8$ $2^{5}$ $2^{2}$ 14 $C_2^2$ $C_2^3$ $C_2^2$ $C_{2}^{3}$ Nilpotent - 2
32.30 $C_2^2.D_4$ $2^{5}$ $2^{2}$ 14 $C_2^2$ $C_2^3$ $C_2^2$ $C_{2}^{3}$ Nilpotent - 2
32.31 $C_4^2:C_2$ $2^{5}$ $2^{2}$ 14 $C_2^2$ $C_2^3$ $C_2^2$ $C_{2}^{3}$ Nilpotent - 2
32.32 $C_4.Q_8$ $2^{5}$ $2^{2}$ 14 $C_2^2$ $C_2^3$ $C_2^2$ $C_{2}^{3}$ Nilpotent - 2
32.33 $C_4^2:C_2$ $2^{5}$ $2^{2}$ 14 $C_2^2$ $C_2^3$ $C_2^2$ $C_{2}^{3}$ Nilpotent - 2
32.34 $C_4:D_4$ $2^{5}$ $2^{2}$ 14 $C_2^2$ $C_2^3$ $C_2^2$ $C_{2}^{3}$ Nilpotent - 2
32.35 $C_4:Q_8$ $2^{5}$ $2^{2}$ 14 $C_2^2$ $C_2^3$ $C_2^2$ $C_{2}^{3}$ Nilpotent - 2
32.36 $C_2^2\times C_8$ $2^{5}$ $2^{3}$ 32 $C_2^2\times C_8$ $C_1$ $C_1$ $C_{2}^{2} \times C_{8}$ Abelian - 3
32.37 $C_2\times \OD_{16}$ $2^{5}$ $2^{3}$ 20 $C_2\times C_4$ $C_2^2$ $C_2$ $C_{2}^{2} \times C_{4}$ Nilpotent - 2
32.38 $\OD_{16}:C_2$ $2^{5}$ $2^{3}$ 20 $C_8$ $C_2^2$ $C_2$ $C_{2}^{2} \times C_{4}$ Nilpotent - 2
32.39 $C_2\times D_8$ $2^{5}$ $2^{3}$ 14 $C_2^2$ $D_4$ $C_4$ $C_{2}^{3}$ Nilpotent - 3
32.40 $C_2\times \SD_{16}$ $2^{5}$ $2^{3}$ 14 $C_2^2$ $D_4$ $C_4$ $C_{2}^{3}$ Nilpotent - 3
32.41 $C_2\times Q_{16}$ $2^{5}$ $2^{3}$ 14 $C_2^2$ $D_4$ $C_4$ $C_{2}^{3}$ Nilpotent - 3
32.42 $D_8:C_2$ $2^{5}$ $2^{3}$ 14 $C_4$ $D_4$ $C_4$ $C_{2}^{3}$ Nilpotent - 3
32.43 $D_8:C_2$ $2^{5}$ $2^{3}$ 11 $C_2$ $C_2\times D_4$ $C_4$ $C_{2}^{3}$ Nilpotent - 3
32.44 $Q_{16}:C_2$ $2^{5}$ $2^{3}$ 11 $C_2$ $C_2\times D_4$ $C_4$ $C_{2}^{3}$ Nilpotent - 3
32.45 $C_2^3\times C_4$ $2^{5}$ $2^{2}$ 32 $C_2^3\times C_4$ $C_1$ $C_1$ $C_{2}^{3} \times C_{4}$ Abelian - 4
32.46 $C_2^2\times D_4$ $2^{5}$ $2^{2}$ 20 $C_2^3$ $C_2^2$ $C_2$ $C_{2}^{4}$ Nilpotent - 2
32.47 $C_2^2\times Q_8$ $2^{5}$ $2^{2}$ 20 $C_2^3$ $C_2^2$ $C_2$ $C_{2}^{4}$ Nilpotent - 2
32.48 $D_4:C_2^2$ $2^{5}$ $2^{2}$ 20 $C_2\times C_4$ $C_2^2$ $C_2$ $C_{2}^{4}$ Nilpotent - 2
32.49 $D_4:C_2^2$ $2^{5}$ $2^{2}$ 17 $C_2$ $C_2^4$ $C_2$ $C_{2}^{4}$ Nilpotent - 2
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