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 25711 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
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
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
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
36.9	$C_3^2:C_4$	$2^{2} \cdot 3^{2}$	$2^{2} \cdot 3$	$-1$	$2$	$4$	$2$	$6$	$38$	$10$	$4$	$C_1$	$C_3^2:C_4$	$C_3^2$	$C_{4}$	$2^{4} \cdot 3^{2}$	$2^{2}$	$6$	$6$	$4$	$4$	$4$	$4$	$4$	$4$	Solvable - 2
36.11	$C_3\times A_4$	$2^{2} \cdot 3^{2}$	$2 \cdot 3$	$-1$	$2$	$4$	$2$	$12$	$30$	$14$	$8$	$C_3$	$A_4$	$C_2^2$	$C_{3}^{2}$	$2^{4} \cdot 3^{2}$	$2^{2} \cdot 3$	$12$	$7$	$3$	$6$	$6$	$3$	$5$	$5$	Solvable - 2
48.3	$C_4^2:C_3$	$2^{4} \cdot 3$	$2^{2} \cdot 3$	$-1$	$2$	$5$	$2$	$8$	$36$	$10$	$4$	$C_1$	$C_4^2:C_3$	$C_4^2$	$C_{3}$	$2^{7} \cdot 3$	$2^{3}$	$12$	$12$	$3$	$6$	$6$	$3$	$6$	$6$	Solvable - 2
48.30	$A_4:C_4$	$2^{4} \cdot 3$	$2^{2} \cdot 3$	$-1$	$3$	$5$	$2$	$10$	$52$	$19$	$7$	$C_2$	$S_4$	$A_4$	$C_{4}$	$2^{4} \cdot 3$	$2$	$12$	$8$	$3$	$6$	$6$	$3$	$5$	$5$	Solvable - 3
48.31	$C_4\times A_4$	$2^{4} \cdot 3$	$2^{2} \cdot 3$	$-1$	$2$	$5$	$2$	$16$	$42$	$19$	$9$	$C_4$	$A_4$	$C_2^2$	$C_{12}$	$2^{4} \cdot 3$	$2^{2}$	$12$	$8$	$3$	$6$	$6$	$3$	$5$	$5$	Solvable - 2
48.48	$C_2\times S_4$	$2^{4} \cdot 3$	$2^{2} \cdot 3$	$-1$	$3$	$5$	$2$	$10$	$98$	$33$	$9$	$C_2$	$S_4$	$A_4$	$C_{2}^{2}$	$2^{4} \cdot 3$	$2$	$6$	$6$	$3$	$3$	$3$	$3$	$3$	$3$	Solvable - 3
48.49	$C_2^2\times A_4$	$2^{4} \cdot 3$	$2 \cdot 3$	$-1$	$2$	$5$	$2$	$16$	$92$	$39$	$15$	$C_2^2$	$A_4$	$C_2^2$	$C_{2} \times C_{6}$	$2^{4} \cdot 3^{2}$	$2^{2} \cdot 3$	$12$	$8$	$-$	$-$	$-$	$4$	$4$	$4$	Solvable - 2
48.50	$C_2^2:A_4$	$2^{4} \cdot 3$	$2 \cdot 3$	$-1$	$2$	$5$	$3$	$8$	$104$	$34$	$8$	$C_1$	$C_2^2:A_4$	$C_2^4$	$C_{3}$	$2^{7} \cdot 3^{2} \cdot 5$	$2^{3} \cdot 3 \cdot 5$	$12$	$8$	$-$	$-$	$-$	$6$	$6$	$6$	Solvable - 2
56.11	$F_8$	$2^{3} \cdot 7$	$2 \cdot 7$	$-1$	$2$	$4$	$2$	$8$	$25$	$6$	$3$	$C_1$	$F_8$	$C_2^3$	$C_{7}$	$2^{3} \cdot 3 \cdot 7$	$3$	$8$	$8$	$7$	$7$	$7$	$7$	$7$	$7$	Solvable - 2
60.9	$C_5\times A_4$	$2^{2} \cdot 3 \cdot 5$	$2 \cdot 3 \cdot 5$	$-1$	$2$	$4$	$2$	$20$	$20$	$10$	$6$	$C_5$	$A_4$	$C_2^2$	$C_{15}$	$2^{5} \cdot 3$	$2^{3}$	$20$	$9$	$3$	$6$	$12$	$3$	$5$	$7$	Solvable - 2
72.15	$C_2^2:D_9$	$2^{3} \cdot 3^{2}$	$2^{2} \cdot 3^{2}$	$-1$	$3$	$5$	$2$	$9$	$80$	$18$	$6$	$C_1$	$C_2^2:D_9$	$C_2^2:C_9$	$C_{2}$	$2^{3} \cdot 3^{3}$	$3$	$18$	$13$	$6$	$6$	$6$	$5$	$5$	$6$	Solvable - 3
72.16	$C_2^2:C_{18}$	$2^{3} \cdot 3^{2}$	$2 \cdot 3^{2}$	$-1$	$2$	$5$	$2$	$24$	$42$	$20$	$10$	$C_6$	$A_4$	$C_2^2$	$C_{18}$	$2^{3} \cdot 3^{2}$	$2 \cdot 3$	$18$	$15$	$3$	$6$	$6$	$3$	$5$	$6$	Solvable - 2
72.19	$C_3^2:C_8$	$2^{3} \cdot 3^{2}$	$2^{3} \cdot 3$	$-1$	$2$	$5$	$2$	$12$	$44$	$14$	$6$	$C_2$	$C_3^2:C_4$	$C_3^2$	$C_{8}$	$2^{5} \cdot 3^{2}$	$2^{3}$	$24$	$14$	$4$	$8$	$8$	$4$	$6$	$8$	Solvable - 2
72.39	$F_9$	$2^{3} \cdot 3^{2}$	$2^{3} \cdot 3$	$-1$	$2$	$5$	$2$	$9$	$48$	$10$	$5$	$C_1$	$F_9$	$C_3^2$	$C_{8}$	$2^{4} \cdot 3^{2}$	$2$	$9$	$9$	$8$	$8$	$8$	$8$	$8$	$8$	Solvable - 2
72.40	$\SOPlus(4,2)$	$2^{3} \cdot 3^{2}$	$2^{2} \cdot 3$	$-1$	$3$	$5$	$2$	$9$	$112$	$26$	$7$	$C_1$	$\SOPlus(4,2)$	$C_3:S_3$	$C_{2}^{2}$	$2^{4} \cdot 3^{2}$	$2$	$6$	$6$	$4$	$4$	$4$	$4$	$4$	$4$	Solvable - 3
72.41	$\PSU(3,2)$	$2^{3} \cdot 3^{2}$	$2^{2} \cdot 3$	$-1$	$3$	$5$	$2$	$6$	$68$	$14$	$7$	$C_1$	$\PSU(3,2)$	$C_3:S_3$	$C_{2}^{2}$	$2^{4} \cdot 3^{3}$	$2 \cdot 3$	$9$	$9$	$8$	$8$	$8$	$8$	$8$	$8$	Solvable - 3
72.42	$C_3\times S_4$	$2^{3} \cdot 3^{2}$	$2^{2} \cdot 3$	$-1$	$3$	$5$	$2$	$15$	$70$	$24$	$8$	$C_3$	$S_4$	$A_4$	$C_{6}$	$2^{4} \cdot 3$	$2$	$12$	$7$	$3$	$6$	$6$	$3$	$5$	$5$	Solvable - 3
72.43	$C_3:S_4$	$2^{3} \cdot 3^{2}$	$2^{2} \cdot 3$	$-1$	$3$	$5$	$3$	$9$	$140$	$30$	$9$	$C_1$	$C_3:S_4$	$C_3\times A_4$	$C_{2}$	$2^{4} \cdot 3^{3}$	$2 \cdot 3$	$12$	$7$	$6$	$6$	$6$	$5$	$5$	$5$	Solvable - 3
72.44	$S_3\times A_4$	$2^{3} \cdot 3^{2}$	$2 \cdot 3$	$-1$	$2$	$5$	$2$	$12$	$94$	$26$	$9$	$C_1$	$S_3\times A_4$	$C_2\times C_6$	$C_{6}$	$2^{4} \cdot 3^{2}$	$2$	$12$	$7$	$6$	$6$	$6$	$5$	$5$	$5$	Solvable - 2
72.45	$C_2\times C_3^2:C_4$	$2^{3} \cdot 3^{2}$	$2^{2} \cdot 3$	$-1$	$2$	$5$	$2$	$12$	$108$	$26$	$10$	$C_2$	$C_3^2:C_4$	$C_3^2$	$C_{2} \times C_{4}$	$2^{5} \cdot 3^{2}$	$2^{3}$	$12$	$8$	$4$	$4$	$4$	$4$	$4$	$4$	Solvable - 2
72.47	$C_6\times A_4$	$2^{3} \cdot 3^{2}$	$2 \cdot 3$	$-1$	$2$	$5$	$2$	$24$	$72$	$32$	$16$	$C_6$	$A_4$	$C_2^2$	$C_{3} \times C_{6}$	$2^{4} \cdot 3^{2}$	$2^{2} \cdot 3$	$18$	$9$	$3$	$6$	$6$	$3$	$5$	$5$	Solvable - 2
75.2	$C_5^2:C_3$	$3 \cdot 5^{2}$	$3 \cdot 5$	$-1$	$2$	$3$	$2$	$11$	$34$	$6$	$3$	$C_1$	$C_5^2:C_3$	$C_5^2$	$C_{3}$	$2^{4} \cdot 3 \cdot 5^{2}$	$2^{4}$	$15$	$15$	$3$	$6$	$12$	$3$	$6$	$12$	Solvable - 2
80.49	$C_2^4:C_5$	$2^{4} \cdot 5$	$2 \cdot 5$	$-1$	$2$	$5$	$2$	$8$	$84$	$17$	$3$	$C_1$	$C_2^4:C_5$	$C_2^4$	$C_{5}$	$2^{6} \cdot 3 \cdot 5$	$2^{2} \cdot 3$	$10$	$10$	$5$	$5$	$5$	$5$	$5$	$5$	Solvable - 2
84.10	$C_7\times A_4$	$2^{2} \cdot 3 \cdot 7$	$2 \cdot 3 \cdot 7$	$-1$	$2$	$4$	$2$	$28$	$20$	$10$	$6$	$C_7$	$A_4$	$C_2^2$	$C_{21}$	$2^{4} \cdot 3^{2}$	$2^{2} \cdot 3$	$28$	$11$	$3$	$6$	$18$	$3$	$5$	$9$	Solvable - 2
84.11	$C_7:A_4$	$2^{2} \cdot 3 \cdot 7$	$2 \cdot 3 \cdot 7$	$-1$	$2$	$4$	$2$	$12$	$50$	$10$	$5$	$C_1$	$C_7:A_4$	$C_2\times C_{14}$	$C_{3}$	$2^{3} \cdot 3^{2} \cdot 7$	$2 \cdot 3$	$28$	$11$	$3$	$6$	$18$	$3$	$6$	$9$	Solvable - 2
96.64	$C_4^2:S_3$	$2^{5} \cdot 3$	$2^{3} \cdot 3$	$-1$	$3$	$6$	$2$	$10$	$114$	$24$	$5$	$C_1$	$C_4^2:S_3$	$C_4^2:C_3$	$C_{2}$	$2^{6} \cdot 3$	$2$	$12$	$12$	$3$	$6$	$6$	$3$	$6$	$6$	Solvable - 3
96.65	$A_4:C_8$	$2^{5} \cdot 3$	$2^{3} \cdot 3$	$-1$	$3$	$6$	$2$	$20$	$68$	$26$	$10$	$C_4$	$S_4$	$A_4$	$C_{8}$	$2^{5} \cdot 3$	$2^{2}$	$24$	$12$	$3$	$6$	$12$	$3$	$5$	$7$	Solvable - 3
96.68	$C_4^2:C_6$	$2^{5} \cdot 3$	$2^{2} \cdot 3$	$-1$	$2$	$6$	$2$	$16$	$96$	$28$	$8$	$C_2$	$C_4^2:C_3$	$C_4^2$	$C_{6}$	$2^{7} \cdot 3$	$2^{3}$	$12$	$12$	$3$	$6$	$6$	$3$	$6$	$6$	Solvable - 2
96.70	$C_2^3:A_4$	$2^{5} \cdot 3$	$2^{2} \cdot 3$	$-1$	$2$	$6$	$2$	$10$	$164$	$35$	$7$	$C_1$	$C_2^3:A_4$	$C_2^4$	$C_{6}$	$2^{7} \cdot 3$	$2^{2}$	$8$	$8$	$6$	$6$	$6$	$6$	$6$	$6$	Solvable - 2
96.71	$C_4^2:C_6$	$2^{5} \cdot 3$	$2^{2} \cdot 3$	$-1$	$2$	$6$	$2$	$10$	$84$	$20$	$7$	$C_1$	$C_4^2:C_6$	$C_4^2$	$C_{6}$	$2^{6} \cdot 3$	$2$	$16$	$16$	$6$	$12$	$12$	$6$	$12$	$12$	Solvable - 2
96.72	$C_4^2:C_6$	$2^{5} \cdot 3$	$2^{2} \cdot 3$	$-1$	$2$	$6$	$2$	$10$	$132$	$28$	$7$	$C_1$	$C_4^2:C_6$	$C_4^2$	$C_{6}$	$2^{7} \cdot 3$	$2^{2}$	$12$	$12$	$6$	$6$	$6$	$6$	$6$	$6$	Solvable - 2
96.73	$C_8\times A_4$	$2^{5} \cdot 3$	$2^{3} \cdot 3$	$-1$	$2$	$6$	$2$	$32$	$58$	$26$	$12$	$C_8$	$A_4$	$C_2^2$	$C_{24}$	$2^{5} \cdot 3$	$2^{3}$	$24$	$12$	$3$	$6$	$12$	$3$	$5$	$7$	Solvable - 2
96.185	$A_4:Q_8$	$2^{5} \cdot 3$	$2^{2} \cdot 3$	$-1$	$3$	$6$	$2$	$14$	$126$	$42$	$12$	$C_2$	$C_2\times S_4$	$C_2\times A_4$	$C_{2}^{2}$	$2^{6} \cdot 3$	$2^{2}$	$24$	$12$	$6$	$12$	$12$	$5$	$7$	$7$	Solvable - 3
96.186	$C_4\times S_4$	$2^{5} \cdot 3$	$2^{2} \cdot 3$	$-1$	$3$	$6$	$2$	$20$	$172$	$56$	$14$	$C_4$	$S_4$	$A_4$	$C_{2} \times C_{4}$	$2^{5} \cdot 3$	$2^{2}$	$12$	$8$	$3$	$6$	$6$	$3$	$5$	$5$	Solvable - 3
96.187	$C_4:S_4$	$2^{5} \cdot 3$	$2^{2} \cdot 3$	$-1$	$3$	$6$	$2$	$14$	$218$	$56$	$12$	$C_2$	$C_2\times S_4$	$C_2\times A_4$	$C_{2}^{2}$	$2^{6} \cdot 3$	$2^{2}$	$12$	$8$	$6$	$6$	$6$	$5$	$5$	$5$	Solvable - 3
96.194	$C_2^2.S_4$	$2^{5} \cdot 3$	$2^{2} \cdot 3$	$-1$	$3$	$6$	$2$	$20$	$188$	$63$	$18$	$C_2^2$	$S_4$	$A_4$	$C_{2} \times C_{4}$	$2^{6} \cdot 3$	$2^{3}$	$24$	$10$	$-$	$-$	$-$	$4$	$5$	$5$	Solvable - 3
96.195	$\GL(2,\mathbb{Z}/4)$	$2^{5} \cdot 3$	$2^{2} \cdot 3$	$-1$	$3$	$6$	$2$	$14$	$234$	$62$	$12$	$C_2$	$C_2\times S_4$	$C_2\times A_4$	$C_{2}^{2}$	$2^{5} \cdot 3$	$2$	$12$	$8$	$6$	$6$	$6$	$5$	$5$	$5$	Solvable - 3
96.196	$C_2^3:C_{12}$	$2^{5} \cdot 3$	$2^{2} \cdot 3$	$-1$	$2$	$6$	$2$	$32$	$158$	$66$	$24$	$C_2\times C_4$	$A_4$	$C_2^2$	$C_{2} \times C_{12}$	$2^{6} \cdot 3$	$2^{4}$	$24$	$10$	$-$	$-$	$-$	$4$	$5$	$5$	Solvable - 2
96.197	$D_4\times A_4$	$2^{5} \cdot 3$	$2^{2} \cdot 3$	$-1$	$2$	$6$	$2$	$20$	$208$	$66$	$18$	$C_2$	$C_2^2\times A_4$	$C_2^3$	$C_{2} \times C_{6}$	$2^{6} \cdot 3$	$2^{2}$	$12$	$8$	$6$	$6$	$6$	$5$	$5$	$5$	Solvable - 2
96.199	$Q_8\times A_4$	$2^{5} \cdot 3$	$2^{2} \cdot 3$	$-1$	$2$	$6$	$2$	$20$	$108$	$46$	$18$	$C_2$	$C_2^2\times A_4$	$C_2^3$	$C_{2} \times C_{6}$	$2^{6} \cdot 3^{2}$	$2^{2} \cdot 3$	$24$	$12$	$6$	$12$	$12$	$5$	$7$	$7$	Solvable - 2
96.204	$Q_8:A_4$	$2^{5} \cdot 3$	$2^{2} \cdot 3$	$-1$	$3$	$6$	$3$	$11$	$175$	$46$	$9$	$C_2$	$C_2^2:A_4$	$D_4:C_2^2$	$C_{3}$	$2^{6} \cdot 3^{2}$	$2^{2} \cdot 3$	$8$	$8$	$4$	$4$	$4$	$4$	$4$	$4$	Solvable - 3
96.226	$C_2^2\times S_4$	$2^{5} \cdot 3$	$2^{2} \cdot 3$	$-1$	$3$	$6$	$3$	$20$	$420$	$131$	$26$	$C_2^2$	$S_4$	$A_4$	$C_{2}^{3}$	$2^{6} \cdot 3^{2}$	$2^{3} \cdot 3$	$12$	$8$	$-$	$-$	$-$	?	?	?	Solvable - 3
96.227	$C_2^2:S_4$	$2^{5} \cdot 3$	$2^{2} \cdot 3$	$-1$	$3$	$6$	$2$	$10$	$250$	$56$	$7$	$C_1$	$C_2^2:S_4$	$C_2^2:A_4$	$C_{2}$	$2^{6} \cdot 3^{2}$	$2 \cdot 3$	$8$	$8$	$6$	$6$	$6$	$6$	$6$	$6$	Solvable - 3
96.228	$C_2^3\times A_4$	$2^{5} \cdot 3$	$2 \cdot 3$	$-1$	$2$	$6$	$3$	$32$	$454$	$178$	$48$	$C_2^3$	$A_4$	$C_2^2$	$C_{2}^{2} \times C_{6}$	$2^{6} \cdot 3^{2} \cdot 7$	$2^{4} \cdot 3 \cdot 7$	$24$	$10$	$-$	$-$	$-$	?	?	?	Solvable - 2
96.229	$C_2^3:A_4$	$2^{5} \cdot 3$	$2 \cdot 3$	$-1$	$2$	$6$	$3$	$16$	$448$	$148$	$16$	$C_2$	$C_2^2:A_4$	$C_2^4$	$C_{6}$	$2^{7} \cdot 3^{2} \cdot 5$	$2^{3} \cdot 3 \cdot 5$	$12$	$10$	$-$	$-$	$-$	?	?	?	Solvable - 2
108.3	$C_2^2:C_{27}$	$2^{2} \cdot 3^{3}$	$2 \cdot 3^{3}$	$-1$	$2$	$5$	$2$	$36$	$20$	$11$	$7$	$C_9$	$A_4$	$C_2^2$	$C_{27}$	$2^{3} \cdot 3^{3}$	$2 \cdot 3^{2}$	$54$	$31$	$3$	$6$	$18$	$3$	$5$	$18$	Solvable - 2

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Abstract groups
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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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Results (1-50 of 25711 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.

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