LMFDB - Abstract group search results

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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

Supersolvable	Monomial	Rational

Number of subgroups	Number of normal subgroups	Number of conjugacy classes

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 455903 matches)

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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

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

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