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Order Exponent Nilpotency class Nilpotent
Center Commutator Central quotient Abelianization
Abelian Cyclic Solvable Simple
Perfect Direct product Semidirect product Wreath product
Automorphism group Automorphisms Transitive degree Permutation degree
Minimal degree of $\C$-irrep Minimal degree of $\Q$-irrep Minimal degree of $\C$-rep Minimal degree of $\Q$-rep
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Results (1-50 of 512750 matches)

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Label Name Order Exponent $\card{\mathrm{conj}(G)}$ Center Central quotient Commutator Abelianization Type - length
1.1 $C_1$ $1$ $1$ $1$ $C_1$ $C_1$ $C_1$ $C_1$ Trivial
2.1 $C_2$ $2$ $2$ $2$ $C_2$ $C_1$ $C_1$ $C_{2}$ Cyclic
3.1 $C_3$ $3$ $3$ $3$ $C_3$ $C_1$ $C_1$ $C_{3}$ Cyclic
4.1 $C_4$ $2^{2}$ $2^{2}$ $4$ $C_4$ $C_1$ $C_1$ $C_{4}$ Cyclic
4.2 $C_2^2$ $2^{2}$ $2$ $4$ $C_2^2$ $C_1$ $C_1$ $C_{2}^{2}$ Abelian - 2
5.1 $C_5$ $5$ $5$ $5$ $C_5$ $C_1$ $C_1$ $C_{5}$ Cyclic
6.1 $S_3$ $2 \cdot 3$ $2 \cdot 3$ $3$ $C_1$ $S_3$ $C_3$ $C_{2}$ Solvable - 2
6.2 $C_6$ $2 \cdot 3$ $2 \cdot 3$ $6$ $C_6$ $C_1$ $C_1$ $C_{6}$ Cyclic
7.1 $C_7$ $7$ $7$ $7$ $C_7$ $C_1$ $C_1$ $C_{7}$ Cyclic
8.1 $C_8$ $2^{3}$ $2^{3}$ $8$ $C_8$ $C_1$ $C_1$ $C_{8}$ Cyclic
8.2 $C_2\times C_4$ $2^{3}$ $2^{2}$ $8$ $C_2\times C_4$ $C_1$ $C_1$ $C_{2} \times C_{4}$ Abelian - 2
8.3 $D_4$ $2^{3}$ $2^{2}$ $5$ $C_2$ $C_2^2$ $C_2$ $C_{2}^{2}$ Nilpotent - 2
8.4 $Q_8$ $2^{3}$ $2^{2}$ $5$ $C_2$ $C_2^2$ $C_2$ $C_{2}^{2}$ Nilpotent - 2
8.5 $C_2^3$ $2^{3}$ $2$ $8$ $C_2^3$ $C_1$ $C_1$ $C_{2}^{3}$ Abelian - 3
9.1 $C_9$ $3^{2}$ $3^{2}$ $9$ $C_9$ $C_1$ $C_1$ $C_{9}$ Cyclic
9.2 $C_3^2$ $3^{2}$ $3$ $9$ $C_3^2$ $C_1$ $C_1$ $C_{3}^{2}$ Abelian - 2
10.1 $D_5$ $2 \cdot 5$ $2 \cdot 5$ $4$ $C_1$ $D_5$ $C_5$ $C_{2}$ Solvable - 2
10.2 $C_{10}$ $2 \cdot 5$ $2 \cdot 5$ $10$ $C_{10}$ $C_1$ $C_1$ $C_{10}$ Cyclic
11.1 $C_{11}$ $11$ $11$ $11$ $C_{11}$ $C_1$ $C_1$ $C_{11}$ Cyclic
12.1 $C_3:C_4$ $2^{2} \cdot 3$ $2^{2} \cdot 3$ $6$ $C_2$ $S_3$ $C_3$ $C_{4}$ Solvable - 2
12.2 $C_{12}$ $2^{2} \cdot 3$ $2^{2} \cdot 3$ $12$ $C_{12}$ $C_1$ $C_1$ $C_{12}$ Cyclic
12.3 $A_4$ $2^{2} \cdot 3$ $2 \cdot 3$ $4$ $C_1$ $A_4$ $C_2^2$ $C_{3}$ Solvable - 2
12.4 $D_6$ $2^{2} \cdot 3$ $2 \cdot 3$ $6$ $C_2$ $S_3$ $C_3$ $C_{2}^{2}$ Solvable - 2
12.5 $C_2\times C_6$ $2^{2} \cdot 3$ $2 \cdot 3$ $12$ $C_2\times C_6$ $C_1$ $C_1$ $C_{2} \times C_{6}$ Abelian - 2
13.1 $C_{13}$ $13$ $13$ $13$ $C_{13}$ $C_1$ $C_1$ $C_{13}$ Cyclic
14.1 $D_7$ $2 \cdot 7$ $2 \cdot 7$ $5$ $C_1$ $D_7$ $C_7$ $C_{2}$ Solvable - 2
14.2 $C_{14}$ $2 \cdot 7$ $2 \cdot 7$ $14$ $C_{14}$ $C_1$ $C_1$ $C_{14}$ Cyclic
15.1 $C_{15}$ $3 \cdot 5$ $3 \cdot 5$ $15$ $C_{15}$ $C_1$ $C_1$ $C_{15}$ Cyclic
16.1 $C_{16}$ $2^{4}$ $2^{4}$ $16$ $C_{16}$ $C_1$ $C_1$ $C_{16}$ Cyclic
16.2 $C_4^2$ $2^{4}$ $2^{2}$ $16$ $C_4^2$ $C_1$ $C_1$ $C_{4}^{2}$ Abelian - 2
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
16.4 $C_4:C_4$ $2^{4}$ $2^{2}$ $10$ $C_2^2$ $C_2^2$ $C_2$ $C_{2} \times C_{4}$ Nilpotent - 2
16.5 $C_2\times C_8$ $2^{4}$ $2^{3}$ $16$ $C_2\times C_8$ $C_1$ $C_1$ $C_{2} \times C_{8}$ Abelian - 2
16.6 $\OD_{16}$ $2^{4}$ $2^{3}$ $10$ $C_4$ $C_2^2$ $C_2$ $C_{2} \times C_{4}$ Nilpotent - 2
16.7 $D_8$ $2^{4}$ $2^{3}$ $7$ $C_2$ $D_4$ $C_4$ $C_{2}^{2}$ Nilpotent - 3
16.8 $\SD_{16}$ $2^{4}$ $2^{3}$ $7$ $C_2$ $D_4$ $C_4$ $C_{2}^{2}$ Nilpotent - 3
16.9 $Q_{16}$ $2^{4}$ $2^{3}$ $7$ $C_2$ $D_4$ $C_4$ $C_{2}^{2}$ Nilpotent - 3
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
17.1 $C_{17}$ $17$ $17$ $17$ $C_{17}$ $C_1$ $C_1$ $C_{17}$ Cyclic
18.1 $D_9$ $2 \cdot 3^{2}$ $2 \cdot 3^{2}$ $6$ $C_1$ $D_9$ $C_9$ $C_{2}$ Solvable - 2
18.2 $C_{18}$ $2 \cdot 3^{2}$ $2 \cdot 3^{2}$ $18$ $C_{18}$ $C_1$ $C_1$ $C_{18}$ Cyclic
18.3 $C_3\times S_3$ $2 \cdot 3^{2}$ $2 \cdot 3$ $9$ $C_3$ $S_3$ $C_3$ $C_{6}$ Solvable - 2
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
18.5 $C_3\times C_6$ $2 \cdot 3^{2}$ $2 \cdot 3$ $18$ $C_3\times C_6$ $C_1$ $C_1$ $C_{3} \times C_{6}$ Abelian - 2
19.1 $C_{19}$ $19$ $19$ $19$ $C_{19}$ $C_1$ $C_1$ $C_{19}$ Cyclic
20.1 $C_5:C_4$ $2^{2} \cdot 5$ $2^{2} \cdot 5$ $8$ $C_2$ $D_5$ $C_5$ $C_{4}$ Solvable - 2
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