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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
Outer aut. group Outer automorphisms Metabelian Metacyclic
Almost simple Quasisimple A-group Z-group
Frattini subgroup Derived length Rank Schur multiplier
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Number of characteristic subgroups Number of autjugacy classes Number of divisions
Order statistics Order factorization Commutator length
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Results (1-50 of 6519 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
6.2 $C_6$ $2 \cdot 3$ $2 \cdot 3$ $6$ $C_6$ $C_1$ $C_1$ $C_{6}$ 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
8.5 $C_2^3$ $2^{3}$ $2$ $8$ $C_2^3$ $C_1$ $C_1$ $C_{2}^{3}$ 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
10.2 $C_{10}$ $2 \cdot 5$ $2 \cdot 5$ $10$ $C_{10}$ $C_1$ $C_1$ $C_{10}$ Cyclic
11.1 $C_{11}$ $11$ $11$ $11$ $C_{11}$ $C_1$ $C_1$ $C_{11}$ Cyclic
12.2 $C_{12}$ $2^{2} \cdot 3$ $2^{2} \cdot 3$ $12$ $C_{12}$ $C_1$ $C_1$ $C_{12}$ Cyclic
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
14.2 $C_{14}$ $2 \cdot 7$ $2 \cdot 7$ $14$ $C_{14}$ $C_1$ $C_1$ $C_{14}$ 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
16.5 $C_2\times C_8$ $2^{4}$ $2^{3}$ $16$ $C_2\times C_8$ $C_1$ $C_1$ $C_{2} \times C_{8}$ Abelian - 2
16.10 $C_2^2\times C_4$ $2^{4}$ $2^{2}$ $16$ $C_2^2\times C_4$ $C_1$ $C_1$ $C_{2}^{2} \times C_{4}$ Abelian - 3
16.14 $C_2^4$ $2^{4}$ $2$ $16$ $C_2^4$ $C_1$ $C_1$ $C_{2}^{4}$ Abelian - 4
17.1 $C_{17}$ $17$ $17$ $17$ $C_{17}$ $C_1$ $C_1$ $C_{17}$ Cyclic
18.2 $C_{18}$ $2 \cdot 3^{2}$ $2 \cdot 3^{2}$ $18$ $C_{18}$ $C_1$ $C_1$ $C_{18}$ Cyclic
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
20.2 $C_{20}$ $2^{2} \cdot 5$ $2^{2} \cdot 5$ $20$ $C_{20}$ $C_1$ $C_1$ $C_{20}$ Cyclic
20.5 $C_2\times C_{10}$ $2^{2} \cdot 5$ $2 \cdot 5$ $20$ $C_2\times C_{10}$ $C_1$ $C_1$ $C_{2} \times C_{10}$ Abelian - 2
21.2 $C_{21}$ $3 \cdot 7$ $3 \cdot 7$ $21$ $C_{21}$ $C_1$ $C_1$ $C_{21}$ Cyclic
22.2 $C_{22}$ $2 \cdot 11$ $2 \cdot 11$ $22$ $C_{22}$ $C_1$ $C_1$ $C_{22}$ Cyclic
23.1 $C_{23}$ $23$ $23$ $23$ $C_{23}$ $C_1$ $C_1$ $C_{23}$ Cyclic
24.2 $C_{24}$ $2^{3} \cdot 3$ $2^{3} \cdot 3$ $24$ $C_{24}$ $C_1$ $C_1$ $C_{24}$ Cyclic
24.9 $C_2\times C_{12}$ $2^{3} \cdot 3$ $2^{2} \cdot 3$ $24$ $C_2\times C_{12}$ $C_1$ $C_1$ $C_{2} \times C_{12}$ Abelian - 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
25.1 $C_{25}$ $5^{2}$ $5^{2}$ $25$ $C_{25}$ $C_1$ $C_1$ $C_{25}$ Cyclic
25.2 $C_5^2$ $5^{2}$ $5$ $25$ $C_5^2$ $C_1$ $C_1$ $C_{5}^{2}$ Abelian - 2
26.2 $C_{26}$ $2 \cdot 13$ $2 \cdot 13$ $26$ $C_{26}$ $C_1$ $C_1$ $C_{26}$ Cyclic
27.1 $C_{27}$ $3^{3}$ $3^{3}$ $27$ $C_{27}$ $C_1$ $C_1$ $C_{27}$ Cyclic
27.2 $C_3\times C_9$ $3^{3}$ $3^{2}$ $27$ $C_3\times C_9$ $C_1$ $C_1$ $C_{3} \times C_{9}$ Abelian - 2
27.5 $C_3^3$ $3^{3}$ $3$ $27$ $C_3^3$ $C_1$ $C_1$ $C_{3}^{3}$ Abelian - 3
28.2 $C_{28}$ $2^{2} \cdot 7$ $2^{2} \cdot 7$ $28$ $C_{28}$ $C_1$ $C_1$ $C_{28}$ Cyclic
28.4 $C_2\times C_{14}$ $2^{2} \cdot 7$ $2 \cdot 7$ $28$ $C_2\times C_{14}$ $C_1$ $C_1$ $C_{2} \times C_{14}$ Abelian - 2
29.1 $C_{29}$ $29$ $29$ $29$ $C_{29}$ $C_1$ $C_1$ $C_{29}$ Cyclic
30.4 $C_{30}$ $2 \cdot 3 \cdot 5$ $2 \cdot 3 \cdot 5$ $30$ $C_{30}$ $C_1$ $C_1$ $C_{30}$ Cyclic
31.1 $C_{31}$ $31$ $31$ $31$ $C_{31}$ $C_1$ $C_1$ $C_{31}$ Cyclic
32.1 $C_{32}$ $2^{5}$ $2^{5}$ $32$ $C_{32}$ $C_1$ $C_1$ $C_{32}$ Cyclic
32.3 $C_4\times C_8$ $2^{5}$ $2^{3}$ $32$ $C_4\times C_8$ $C_1$ $C_1$ $C_{4} \times C_{8}$ Abelian - 2
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