↑

Learn more

Refine search

Order Exponent Nilpotency class Nilpotent
Center Commutator Central quotient Abelianization
Abelian Cyclic Solvable Simple
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
Order statistics Family Order factorization Commutator length
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
Almost simple Quasisimple A-group Z-group
Frattini subgroup Derived length Rank Schur multiplier
Supersolvable Monomial Rational
Num. of char. subgroups Num. of aut. classes Number of divisions
Sort order Select

Results (1-50 of 512750 matches)

Next   displayed columns for results
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
Next   displayed columns for results