Note: Search results may be incomplete due to uncomputed quantities: Rational (11543 objects)

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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 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
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Results (1-50 of 6605 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
4.2 $C_2^2$ $2^{2}$ $2$ 4 $C_2^2$ $C_1$ $C_1$ $C_{2}^{2}$ Abelian - 2
6.1 $S_3$ $2 \cdot 3$ $2 \cdot 3$ 3 $C_1$ $S_3$ $C_3$ $C_{2}$ Solvable - 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
12.4 $D_6$ $2^{2} \cdot 3$ $2 \cdot 3$ 6 $C_2$ $S_3$ $C_3$ $C_{2}^{2}$ Solvable - 2
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
16.14 $C_2^4$ $2^{4}$ $2$ 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
24.12 $S_4$ $2^{3} \cdot 3$ $2^{2} \cdot 3$ 5 $C_1$ $S_4$ $A_4$ $C_{2}$ Solvable - 3
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
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
32.34 $C_4:D_4$ $2^{5}$ $2^{2}$ 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
32.43 $D_8:C_2$ $2^{5}$ $2^{3}$ 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
32.46 $C_2^2\times D_4$ $2^{5}$ $2^{2}$ 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
32.49 $D_4:C_2^2$ $2^{5}$ $2^{2}$ 17 $C_2$ $C_2^4$ $C_2$ $C_{2}^{4}$ Nilpotent - 2
32.50 $C_4.C_2^3$ $2^{5}$ $2^{2}$ 17 $C_2$ $C_2^4$ $C_2$ $C_{2}^{4}$ Nilpotent - 2
32.51 $C_2^5$ $2^{5}$ $2$ 32 $C_2^5$ $C_1$ $C_1$ $C_{2}^{5}$ Abelian - 5
36.10 $S_3^2$ $2^{2} \cdot 3^{2}$ $2 \cdot 3$ 9 $C_1$ $S_3^2$ $C_3^2$ $C_{2}^{2}$ Solvable - 2
36.13 $C_6:S_3$ $2^{2} \cdot 3^{2}$ $2 \cdot 3$ 12 $C_2$ $C_3:S_3$ $C_3^2$ $C_{2}^{2}$ Solvable - 2
48.38 $S_3\times D_4$ $2^{4} \cdot 3$ $2^{2} \cdot 3$ 15 $C_2$ $C_2\times D_6$ $C_6$ $C_{2}^{3}$ Solvable - 2
48.40 $S_3\times Q_8$ $2^{4} \cdot 3$ $2^{2} \cdot 3$ 15 $C_2$ $C_2\times D_6$ $C_6$ $C_{2}^{3}$ Solvable - 2
48.48 $C_2\times S_4$ $2^{4} \cdot 3$ $2^{2} \cdot 3$ 10 $C_2$ $S_4$ $A_4$ $C_{2}^{2}$ Solvable - 3
48.51 $C_2^2\times D_6$ $2^{4} \cdot 3$ $2 \cdot 3$ 24 $C_2^3$ $S_3$ $C_3$ $C_{2}^{4}$ Solvable - 2
54.14 $C_3^2:S_3$ $2 \cdot 3^{3}$ $2 \cdot 3$ 15 $C_1$ $C_3^2:S_3$ $C_3^3$ $C_{2}$ Solvable - 2
64.134 $D_4:D_4$ $2^{6}$ $2^{3}$ 16 $C_2$ $C_2^2\wr C_2$ $C_2\times C_4$ $C_{2}^{3}$ Nilpotent - 3
64.137 $D_4.D_4$ $2^{6}$ $2^{3}$ 16 $C_2$ $C_2^2\wr C_2$ $C_2\times C_4$ $C_{2}^{3}$ Nilpotent - 3
64.138 $C_2\wr C_2^2$ $2^{6}$ $2^{2}$ 16 $C_2$ $C_2^2\wr C_2$ $C_2^3$ $C_{2}^{3}$ Nilpotent - 3
64.177 $C_8:D_4$ $2^{6}$ $2^{3}$ 16 $C_2^2$ $C_2\times D_4$ $C_2\times C_4$ $C_{2}^{3}$ Nilpotent - 3
64.178 $C_8.D_4$ $2^{6}$ $2^{3}$ 16 $C_2^2$ $C_2\times D_4$ $C_2\times C_4$ $C_{2}^{3}$ Nilpotent - 3
64.182 $C_8:Q_8$ $2^{6}$ $2^{3}$ 16 $C_2^2$ $C_2\times D_4$ $C_2\times C_4$ $C_{2}^{3}$ Nilpotent - 3
64.202 $C_2^3:D_4$ $2^{6}$ $2^{2}$ 28 $C_2^3$ $C_2^3$ $C_2^2$ $C_{2}^{4}$ Nilpotent - 2
64.211 $C_4^2:C_2^2$ $2^{6}$ $2^{2}$ 28 $C_2^3$ $C_2^3$ $C_2^2$ $C_{2}^{4}$ Nilpotent - 2
64.212 $C_4^2.C_2^2$ $2^{6}$ $2^{2}$ 28 $C_2^3$ $C_2^3$ $C_2^2$ $C_{2}^{4}$ Nilpotent - 2
64.215 $C_2^3:D_4$ $2^{6}$ $2^{2}$ 22 $C_2^2$ $C_2^4$ $C_2^2$ $C_{2}^{4}$ Nilpotent - 2
64.216 $C_4^2:C_2^2$ $2^{6}$ $2^{2}$ 22 $C_2^2$ $C_2^4$ $C_2^2$ $C_{2}^{4}$ Nilpotent - 2
64.217 $C_4^2.C_2^2$ $2^{6}$ $2^{2}$ 22 $C_2^2$ $C_2^4$ $C_2^2$ $C_{2}^{4}$ Nilpotent - 2
64.218 $C_2^2.C_2^4$ $2^{6}$ $2^{2}$ 22 $C_2^2$ $C_2^4$ $C_2^2$ $C_{2}^{4}$ Nilpotent - 2
64.224 $C_2^3:Q_8$ $2^{6}$ $2^{2}$ 22 $C_2^2$ $C_2^4$ $C_2^2$ $C_{2}^{4}$ Nilpotent - 2
64.225 $C_4^2.C_2^2$ $2^{6}$ $2^{2}$ 22 $C_2^2$ $C_2^4$ $C_2^2$ $C_{2}^{4}$ Nilpotent - 2
64.226 $D_4^2$ $2^{6}$ $2^{2}$ 25 $C_2^2$ $C_2^4$ $C_2^2$ $C_{2}^{4}$ Nilpotent - 2
64.230 $D_4\times Q_8$ $2^{6}$ $2^{2}$ 25 $C_2^2$ $C_2^4$ $C_2^2$ $C_{2}^{4}$ Nilpotent - 2
64.239 $Q_8^2$ $2^{6}$ $2^{2}$ 25 $C_2^2$ $C_2^4$ $C_2^2$ $C_{2}^{4}$ Nilpotent - 2
64.241 $C_4^2:C_2^2$ $2^{6}$ $2^{2}$ 19 $C_2^2$ $C_2^4$ $C_2^2$ $C_{2}^{4}$ Nilpotent - 2
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