LMFDB - Abstract group search results

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 455903 matches)

Next Download displayed columns for results to

Label	Name	Order	Exponent	Nilp. class	Der. length	Comp. length	Rank	$\card{\mathrm{conj}(G)}$	Subgroups	Subgroup classes	Normal subgroups	Center	Central quotient	Commutator	Abelianization	$\card{\mathrm{Aut}(G)}$	$\card{\mathrm{Out}(G)}$	Tr. deg	Perm. deg	$\C$-irrep deg	$\R$-irrep deg	$\Q$-irrep deg	$\C$-rep deg	$\R$-rep deg	$\Q$-rep deg	Type - length
6.1	$S_3$	$2 \cdot 3$	$2 \cdot 3$	$-1$	$2$	$2$	$2$	3	$6$	$4$	$3$	$C_1$	$S_3$	$C_3$	$C_{2}$	$2 \cdot 3$	$1$	$3$	$3$	$2$	$2$	$2$	$2$	$2$	$2$	Solvable - 2
10.1	$D_5$	$2 \cdot 5$	$2 \cdot 5$	$-1$	$2$	$2$	$2$	4	$8$	$4$	$3$	$C_1$	$D_5$	$C_5$	$C_{2}$	$2^{2} \cdot 5$	$2$	$5$	$5$	$2$	$2$	$4$	$2$	$2$	$4$	Solvable - 2
12.1	$C_3:C_4$	$2^{2} \cdot 3$	$2^{2} \cdot 3$	$-1$	$2$	$3$	$2$	6	$8$	$6$	$5$	$C_2$	$S_3$	$C_3$	$C_{4}$	$2^{2} \cdot 3$	$2$	$12$	$7$	$2$	$4$	$4$	$2$	$4$	$4$	Solvable - 2
12.3	$A_4$	$2^{2} \cdot 3$	$2 \cdot 3$	$-1$	$2$	$3$	$2$	4	$10$	$5$	$3$	$C_1$	$A_4$	$C_2^2$	$C_{3}$	$2^{3} \cdot 3$	$2$	$4$	$4$	$3$	$3$	$3$	$3$	$3$	$3$	Solvable - 2
12.4	$D_6$	$2^{2} \cdot 3$	$2 \cdot 3$	$-1$	$2$	$3$	$2$	6	$16$	$10$	$7$	$C_2$	$S_3$	$C_3$	$C_{2}^{2}$	$2^{2} \cdot 3$	$2$	$6$	$5$	$2$	$2$	$2$	$2$	$2$	$2$	Solvable - 2
14.1	$D_7$	$2 \cdot 7$	$2 \cdot 7$	$-1$	$2$	$2$	$2$	5	$10$	$4$	$3$	$C_1$	$D_7$	$C_7$	$C_{2}$	$2 \cdot 3 \cdot 7$	$3$	$7$	$7$	$2$	$2$	$6$	$2$	$2$	$6$	Solvable - 2
18.1	$D_9$	$2 \cdot 3^{2}$	$2 \cdot 3^{2}$	$-1$	$2$	$3$	$2$	6	$16$	$6$	$4$	$C_1$	$D_9$	$C_9$	$C_{2}$	$2 \cdot 3^{3}$	$3$	$9$	$9$	$2$	$2$	$6$	$2$	$2$	$6$	Solvable - 2
18.3	$C_3\times S_3$	$2 \cdot 3^{2}$	$2 \cdot 3$	$-1$	$2$	$3$	$2$	9	$14$	$9$	$6$	$C_3$	$S_3$	$C_3$	$C_{6}$	$2^{2} \cdot 3$	$2$	$6$	$6$	$2$	$4$	$4$	$2$	$4$	$4$	Solvable - 2
18.4	$C_3:S_3$	$2 \cdot 3^{2}$	$2 \cdot 3$	$-1$	$2$	$3$	$3$	6	$28$	$12$	$7$	$C_1$	$C_3:S_3$	$C_3^2$	$C_{2}$	$2^{4} \cdot 3^{3}$	$2^{3} \cdot 3$	$9$	$6$	$-$	$-$	$-$	$4$	$4$	$4$	Solvable - 2
20.1	$C_5:C_4$	$2^{2} \cdot 5$	$2^{2} \cdot 5$	$-1$	$2$	$3$	$2$	8	$10$	$6$	$5$	$C_2$	$D_5$	$C_5$	$C_{4}$	$2^{3} \cdot 5$	$2^{2}$	$20$	$9$	$2$	$4$	$8$	$2$	$4$	$6$	Solvable - 2
20.3	$F_5$	$2^{2} \cdot 5$	$2^{2} \cdot 5$	$-1$	$2$	$3$	$2$	5	$14$	$6$	$4$	$C_1$	$F_5$	$C_5$	$C_{4}$	$2^{2} \cdot 5$	$1$	$5$	$5$	$4$	$4$	$4$	$4$	$4$	$4$	Solvable - 2
20.4	$D_{10}$	$2^{2} \cdot 5$	$2 \cdot 5$	$-1$	$2$	$3$	$2$	8	$22$	$10$	$7$	$C_2$	$D_5$	$C_5$	$C_{2}^{2}$	$2^{3} \cdot 5$	$2^{2}$	$10$	$7$	$2$	$2$	$4$	$2$	$2$	$4$	Solvable - 2
21.1	$C_7:C_3$	$3 \cdot 7$	$3 \cdot 7$	$-1$	$2$	$2$	$2$	5	$10$	$4$	$3$	$C_1$	$C_7:C_3$	$C_7$	$C_{3}$	$2 \cdot 3 \cdot 7$	$2$	$7$	$7$	$3$	$6$	$6$	$3$	$6$	$6$	Solvable - 2
22.1	$D_{11}$	$2 \cdot 11$	$2 \cdot 11$	$-1$	$2$	$2$	$2$	7	$14$	$4$	$3$	$C_1$	$D_{11}$	$C_{11}$	$C_{2}$	$2 \cdot 5 \cdot 11$	$5$	$11$	$11$	$2$	$2$	$10$	$2$	$2$	$10$	Solvable - 2
24.1	$C_3:C_8$	$2^{3} \cdot 3$	$2^{3} \cdot 3$	$-1$	$2$	$4$	$2$	12	$10$	$8$	$7$	$C_4$	$S_3$	$C_3$	$C_{8}$	$2^{3} \cdot 3$	$2^{2}$	$24$	$11$	$2$	$4$	$4$	$2$	$4$	$4$	Solvable - 2
24.3	$\SL(2,3)$	$2^{3} \cdot 3$	$2^{2} \cdot 3$	$-1$	$3$	$4$	$2$	7	$15$	$7$	$4$	$C_2$	$A_4$	$Q_8$	$C_{3}$	$2^{3} \cdot 3$	$2$	$8$	$8$	$2$	$4$	$4$	$2$	$4$	$4$	Solvable - 3
24.4	$C_3:Q_8$	$2^{3} \cdot 3$	$2^{2} \cdot 3$	$-1$	$2$	$4$	$2$	9	$18$	$12$	$9$	$C_2$	$D_6$	$C_6$	$C_{2}^{2}$	$2^{4} \cdot 3$	$2^{2}$	$24$	$11$	$2$	$4$	$8$	$2$	$4$	$6$	Solvable - 2
24.5	$C_4\times S_3$	$2^{3} \cdot 3$	$2^{2} \cdot 3$	$-1$	$2$	$4$	$2$	12	$26$	$16$	$11$	$C_4$	$S_3$	$C_3$	$C_{2} \times C_{4}$	$2^{3} \cdot 3$	$2^{2}$	$12$	$7$	$2$	$4$	$4$	$2$	$4$	$4$	Solvable - 2
24.6	$D_{12}$	$2^{3} \cdot 3$	$2^{2} \cdot 3$	$-1$	$2$	$4$	$2$	9	$34$	$16$	$9$	$C_2$	$D_6$	$C_6$	$C_{2}^{2}$	$2^{4} \cdot 3$	$2^{2}$	$12$	$7$	$2$	$2$	$4$	$2$	$2$	$4$	Solvable - 2
24.7	$C_6:C_4$	$2^{3} \cdot 3$	$2^{2} \cdot 3$	$-1$	$2$	$4$	$2$	12	$22$	$16$	$13$	$C_2^2$	$S_3$	$C_3$	$C_{2} \times C_{4}$	$2^{4} \cdot 3$	$2^{3}$	$24$	$9$	$-$	$-$	$-$	$3$	$4$	$4$	Solvable - 2
24.8	$C_3:D_4$	$2^{3} \cdot 3$	$2^{2} \cdot 3$	$-1$	$2$	$4$	$2$	9	$30$	$16$	$9$	$C_2$	$D_6$	$C_6$	$C_{2}^{2}$	$2^{3} \cdot 3$	$2$	$12$	$7$	$2$	$4$	$4$	$2$	$4$	$4$	Solvable - 2
24.12	$S_4$	$2^{3} \cdot 3$	$2^{2} \cdot 3$	$-1$	$3$	$4$	$2$	5	$30$	$11$	$4$	$C_1$	$S_4$	$A_4$	$C_{2}$	$2^{3} \cdot 3$	$1$	$4$	$4$	$3$	$3$	$3$	$3$	$3$	$3$	Solvable - 3
24.13	$C_2\times A_4$	$2^{3} \cdot 3$	$2 \cdot 3$	$-1$	$2$	$4$	$2$	8	$26$	$12$	$6$	$C_2$	$A_4$	$C_2^2$	$C_{6}$	$2^{3} \cdot 3$	$2$	$6$	$6$	$3$	$3$	$3$	$3$	$3$	$3$	Solvable - 2
24.14	$C_2\times D_6$	$2^{3} \cdot 3$	$2 \cdot 3$	$-1$	$2$	$4$	$3$	12	$54$	$32$	$21$	$C_2^2$	$S_3$	$C_3$	$C_{2}^{3}$	$2^{4} \cdot 3^{2}$	$2^{3} \cdot 3$	$12$	$7$	$-$	$-$	$-$	$3$	$3$	$3$	Solvable - 2
26.1	$D_{13}$	$2 \cdot 13$	$2 \cdot 13$	$-1$	$2$	$2$	$2$	8	$16$	$4$	$3$	$C_1$	$D_{13}$	$C_{13}$	$C_{2}$	$2^{2} \cdot 3 \cdot 13$	$2 \cdot 3$	$13$	$13$	$2$	$2$	$12$	$2$	$2$	$12$	Solvable - 2
28.1	$C_7:C_4$	$2^{2} \cdot 7$	$2^{2} \cdot 7$	$-1$	$2$	$3$	$2$	10	$12$	$6$	$5$	$C_2$	$D_7$	$C_7$	$C_{4}$	$2^{2} \cdot 3 \cdot 7$	$2 \cdot 3$	$28$	$11$	$2$	$4$	$12$	$2$	$4$	$8$	Solvable - 2
28.3	$D_{14}$	$2^{2} \cdot 7$	$2 \cdot 7$	$-1$	$2$	$3$	$2$	10	$28$	$10$	$7$	$C_2$	$D_7$	$C_7$	$C_{2}^{2}$	$2^{2} \cdot 3 \cdot 7$	$2 \cdot 3$	$14$	$9$	$2$	$2$	$6$	$2$	$2$	$6$	Solvable - 2
30.1	$C_5\times S_3$	$2 \cdot 3 \cdot 5$	$2 \cdot 3 \cdot 5$	$-1$	$2$	$3$	$2$	15	$12$	$8$	$6$	$C_5$	$S_3$	$C_3$	$C_{10}$	$2^{3} \cdot 3$	$2^{2}$	$15$	$8$	$2$	$4$	$8$	$2$	$4$	$6$	Solvable - 2
30.2	$C_3\times D_5$	$2 \cdot 3 \cdot 5$	$2 \cdot 3 \cdot 5$	$-1$	$2$	$3$	$2$	12	$16$	$8$	$6$	$C_3$	$D_5$	$C_5$	$C_{6}$	$2^{3} \cdot 5$	$2^{2}$	$15$	$8$	$2$	$4$	$8$	$2$	$4$	$6$	Solvable - 2
30.3	$D_{15}$	$2 \cdot 3 \cdot 5$	$2 \cdot 3 \cdot 5$	$-1$	$2$	$3$	$2$	9	$28$	$8$	$5$	$C_1$	$D_{15}$	$C_{15}$	$C_{2}$	$2^{3} \cdot 3 \cdot 5$	$2^{2}$	$15$	$8$	$2$	$2$	$8$	$2$	$2$	$6$	Solvable - 2
34.1	$D_{17}$	$2 \cdot 17$	$2 \cdot 17$	$-1$	$2$	$2$	$2$	10	$20$	$4$	$3$	$C_1$	$D_{17}$	$C_{17}$	$C_{2}$	$2^{4} \cdot 17$	$2^{3}$	$17$	$17$	$2$	$2$	$16$	$2$	$2$	$16$	Solvable - 2
36.1	$C_9:C_4$	$2^{2} \cdot 3^{2}$	$2^{2} \cdot 3^{2}$	$-1$	$2$	$4$	$2$	12	$19$	$9$	$7$	$C_2$	$D_9$	$C_9$	$C_{4}$	$2^{2} \cdot 3^{3}$	$2 \cdot 3$	$36$	$13$	$2$	$4$	$12$	$2$	$4$	$8$	Solvable - 2
36.3	$C_2^2:C_9$	$2^{2} \cdot 3^{2}$	$2 \cdot 3^{2}$	$-1$	$2$	$4$	$2$	12	$15$	$8$	$5$	$C_3$	$A_4$	$C_2^2$	$C_{9}$	$2^{3} \cdot 3^{2}$	$2 \cdot 3$	$18$	$13$	$3$	$6$	$6$	$3$	$5$	$6$	Solvable - 2
36.4	$D_{18}$	$2^{2} \cdot 3^{2}$	$2 \cdot 3^{2}$	$-1$	$2$	$4$	$2$	12	$45$	$15$	$9$	$C_2$	$D_9$	$C_9$	$C_{2}^{2}$	$2^{2} \cdot 3^{3}$	$2 \cdot 3$	$18$	$11$	$2$	$2$	$6$	$2$	$2$	$6$	Solvable - 2
36.6	$C_3:C_{12}$	$2^{2} \cdot 3^{2}$	$2^{2} \cdot 3$	$-1$	$2$	$4$	$2$	18	$20$	$14$	$10$	$C_6$	$S_3$	$C_3$	$C_{12}$	$2^{3} \cdot 3$	$2^{2}$	$12$	$10$	$2$	$4$	$4$	$2$	$4$	$4$	Solvable - 2
36.7	$C_3^2:C_4$	$2^{2} \cdot 3^{2}$	$2^{2} \cdot 3$	$-1$	$2$	$4$	$3$	12	$34$	$18$	$13$	$C_2$	$C_3:S_3$	$C_3^2$	$C_{4}$	$2^{5} \cdot 3^{3}$	$2^{4} \cdot 3$	$36$	$10$	$-$	$-$	$-$	$4$	$6$	$6$	Solvable - 2
36.9	$C_3^2:C_4$	$2^{2} \cdot 3^{2}$	$2^{2} \cdot 3$	$-1$	$2$	$4$	$2$	6	$38$	$10$	$4$	$C_1$	$C_3^2:C_4$	$C_3^2$	$C_{4}$	$2^{4} \cdot 3^{2}$	$2^{2}$	$6$	$6$	$4$	$4$	$4$	$4$	$4$	$4$	Solvable - 2
36.10	$S_3^2$	$2^{2} \cdot 3^{2}$	$2 \cdot 3$	$-1$	$2$	$4$	$2$	9	$60$	$22$	$10$	$C_1$	$S_3^2$	$C_3^2$	$C_{2}^{2}$	$2^{3} \cdot 3^{2}$	$2$	$6$	$6$	$4$	$4$	$4$	$4$	$4$	$4$	Solvable - 2
36.11	$C_3\times A_4$	$2^{2} \cdot 3^{2}$	$2 \cdot 3$	$-1$	$2$	$4$	$2$	12	$30$	$14$	$8$	$C_3$	$A_4$	$C_2^2$	$C_{3}^{2}$	$2^{4} \cdot 3^{2}$	$2^{2} \cdot 3$	$12$	$7$	$3$	$6$	$6$	$3$	$5$	$5$	Solvable - 2
36.12	$C_6\times S_3$	$2^{2} \cdot 3^{2}$	$2 \cdot 3$	$-1$	$2$	$4$	$2$	18	$36$	$22$	$14$	$C_6$	$S_3$	$C_3$	$C_{2} \times C_{6}$	$2^{3} \cdot 3$	$2^{2}$	$12$	$8$	$2$	$4$	$4$	$2$	$4$	$4$	Solvable - 2
36.13	$C_6:S_3$	$2^{2} \cdot 3^{2}$	$2 \cdot 3$	$-1$	$2$	$4$	$3$	12	$78$	$30$	$15$	$C_2$	$C_3:S_3$	$C_3^2$	$C_{2}^{2}$	$2^{5} \cdot 3^{3}$	$2^{4} \cdot 3$	$18$	$8$	$-$	$-$	$-$	$4$	$4$	$4$	Solvable - 2
38.1	$D_{19}$	$2 \cdot 19$	$2 \cdot 19$	$-1$	$2$	$2$	$2$	11	$22$	$4$	$3$	$C_1$	$D_{19}$	$C_{19}$	$C_{2}$	$2 \cdot 3^{2} \cdot 19$	$3^{2}$	$19$	$19$	$2$	$2$	$18$	$2$	$2$	$18$	Solvable - 2
39.1	$C_{13}:C_3$	$3 \cdot 13$	$3 \cdot 13$	$-1$	$2$	$2$	$2$	7	$16$	$4$	$3$	$C_1$	$C_{13}:C_3$	$C_{13}$	$C_{3}$	$2^{2} \cdot 3 \cdot 13$	$2^{2}$	$13$	$13$	$3$	$6$	$12$	$3$	$6$	$12$	Solvable - 2
40.1	$C_5:C_8$	$2^{3} \cdot 5$	$2^{3} \cdot 5$	$-1$	$2$	$4$	$2$	16	$12$	$8$	$7$	$C_4$	$D_5$	$C_5$	$C_{8}$	$2^{4} \cdot 5$	$2^{3}$	$40$	$13$	$2$	$4$	$16$	$2$	$4$	$8$	Solvable - 2
40.3	$C_5:C_8$	$2^{3} \cdot 5$	$2^{3} \cdot 5$	$-1$	$2$	$4$	$2$	10	$16$	$8$	$6$	$C_2$	$F_5$	$C_5$	$C_{8}$	$2^{3} \cdot 5$	$2$	$40$	$13$	$4$	$8$	$8$	$4$	$6$	$8$	Solvable - 2
40.4	$C_5:Q_8$	$2^{3} \cdot 5$	$2^{2} \cdot 5$	$-1$	$2$	$4$	$2$	13	$24$	$12$	$9$	$C_2$	$D_{10}$	$C_{10}$	$C_{2}^{2}$	$2^{5} \cdot 5$	$2^{3}$	$40$	$13$	$2$	$4$	$16$	$2$	$4$	$8$	Solvable - 2
40.5	$C_4\times D_5$	$2^{3} \cdot 5$	$2^{2} \cdot 5$	$-1$	$2$	$4$	$2$	16	$36$	$16$	$11$	$C_4$	$D_5$	$C_5$	$C_{2} \times C_{4}$	$2^{4} \cdot 5$	$2^{3}$	$20$	$9$	$2$	$4$	$8$	$2$	$4$	$6$	Solvable - 2
40.6	$D_{20}$	$2^{3} \cdot 5$	$2^{2} \cdot 5$	$-1$	$2$	$4$	$2$	13	$48$	$16$	$9$	$C_2$	$D_{10}$	$C_{10}$	$C_{2}^{2}$	$2^{5} \cdot 5$	$2^{3}$	$20$	$9$	$2$	$2$	$8$	$2$	$2$	$6$	Solvable - 2
40.7	$C_{10}:C_4$	$2^{3} \cdot 5$	$2^{2} \cdot 5$	$-1$	$2$	$4$	$2$	16	$28$	$16$	$13$	$C_2^2$	$D_5$	$C_5$	$C_{2} \times C_{4}$	$2^{5} \cdot 5$	$2^{4}$	$40$	$11$	$-$	$-$	$-$	$3$	$4$	$6$	Solvable - 2
40.8	$C_5:D_4$	$2^{3} \cdot 5$	$2^{2} \cdot 5$	$-1$	$2$	$4$	$2$	13	$40$	$16$	$9$	$C_2$	$D_{10}$	$C_{10}$	$C_{2}^{2}$	$2^{4} \cdot 5$	$2^{2}$	$20$	$9$	$2$	$4$	$8$	$2$	$4$	$6$	Solvable - 2

Next Download displayed columns for results to

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Learn more

Refine search

Results (1-50 of 455903 matches)

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Learn more

Refine search Advanced search options

Results (1-50 of 455903 matches)

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Refine search