LMFDB - Abstract group search results

The results below are complete, since the LMFDB contains all groups of order at most 2000 except orders larger than 500 that are multiples of 128

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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 20169 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
384.1	$C_3:C_{128}$	$2^{7} \cdot 3$	$2^{7} \cdot 3$	$-1$	$2$	$8$	$2$	192	$18$	$16$	$15$	$C_{64}$	$S_3$	$C_3$	$C_{128}$	$2^{7} \cdot 3$	$2^{6}$	$384$	$131$	$2$	$4$	$128$	$2$	$4$	$66$	Solvable - 2
384.2	$C_{384}$	$2^{7} \cdot 3$	$2^{7} \cdot 3$	$1$	$1$	$8$	$1$	384	$16$	$16$	$16$	$C_{384}$	$C_1$	$C_1$	$C_{384}$	$2^{7}$	$2^{7}$	$384$	$131$	$1$	$2$	$128$	$1$	$2$	$66$	Cyclic
384.3	$C_4^2.\SL(2,3)$	$2^{7} \cdot 3$	$2^{3} \cdot 3$	$-1$	$3$	$8$	$2$	32	$305$	$47$	$8$	$C_2$	$C_8^2:C_3$	$C_2.C_8^2$	$C_{3}$	$2^{11} \cdot 3$	$2^{5}$	$24$	$24$	$6$	$12$	$12$	$5$	$10$	$12$	Solvable - 3
384.4	$C_2^4.\SL(2,3)$	$2^{7} \cdot 3$	$2^{2} \cdot 3$	$-1$	$3$	$8$	$2$	14	$897$	$91$	$7$	$C_1$	$C_2^4.\SL(2,3)$	$C_2^3.C_4^2$	$C_{3}$	$2^{11} \cdot 3$	$2^{4}$	$16$	$16$	$12$	$12$	$12$	$12$	$12$	$12$	Solvable - 3
384.5	$C_4^2.\SL(2,3)$	$2^{7} \cdot 3$	$2^{3} \cdot 3$	$-1$	$3$	$8$	$2$	14	$281$	$32$	$7$	$C_1$	$C_4^2.\SL(2,3)$	$C_4^2.D_4$	$C_{3}$	$2^{10} \cdot 3$	$2^{3}$	$48$	$48$	$12$	$24$	$24$	$12$	$24$	$24$	Solvable - 3
384.6	$C_4^2.\SL(2,3)$	$2^{7} \cdot 3$	$2^{3} \cdot 3$	$-1$	$3$	$8$	$2$	14	$305$	$36$	$7$	$C_1$	$C_4^2.\SL(2,3)$	$C_4^2.D_4$	$C_{3}$	$2^{11} \cdot 3$	$2^{4}$	$24$	$24$	$12$	$12$	$12$	$12$	$12$	$12$	Solvable - 3
384.7	$S_3\times C_{64}$	$2^{7} \cdot 3$	$2^{6} \cdot 3$	$-1$	$2$	$8$	$2$	192	$66$	$40$	$27$	$C_{64}$	$S_3$	$C_3$	$C_{2} \times C_{64}$	$2^{7} \cdot 3$	$2^{6}$	$192$	$67$	$2$	$4$	$64$	$2$	$4$	$34$	Solvable - 2
384.8	$C_{192}:C_2$	$2^{7} \cdot 3$	$2^{6} \cdot 3$	$-1$	$2$	$8$	$2$	144	$66$	$38$	$25$	$C_{32}$	$D_6$	$C_6$	$C_{2} \times C_{32}$	$2^{7} \cdot 3$	$2^{5}$	$192$	$67$	$2$	$4$	$64$	$2$	$4$	$34$	Solvable - 2
384.9	$D_{192}$	$2^{7} \cdot 3$	$2^{6} \cdot 3$	$-1$	$2$	$8$	$2$	99	$522$	$40$	$17$	$C_2$	$D_{96}$	$C_{96}$	$C_{2}^{2}$	$2^{12} \cdot 3$	$2^{6}$	$192$	$67$	$2$	$2$	$64$	$2$	$2$	$34$	Solvable - 2
384.10	$C_{192}:C_2$	$2^{7} \cdot 3$	$2^{6} \cdot 3$	$-1$	$2$	$8$	$2$	99	$394$	$38$	$17$	$C_2$	$D_{96}$	$C_{96}$	$C_{2}^{2}$	$2^{11} \cdot 3$	$2^{5}$	$192$	$67$	$2$	$4$	$64$	$2$	$4$	$34$	Solvable - 2
384.11	$C_3:Q_{128}$	$2^{7} \cdot 3$	$2^{6} \cdot 3$	$-1$	$2$	$8$	$2$	99	$266$	$36$	$17$	$C_2$	$D_{96}$	$C_{96}$	$C_{2}^{2}$	$2^{12} \cdot 3$	$2^{6}$	$384$	$131$	$2$	$4$	$128$	$2$	$4$	$66$	Solvable - 2
384.12	$C_{24}:C_{16}$	$2^{7} \cdot 3$	$2^{4} \cdot 3$	$-1$	$2$	$8$	$2$	192	$134$	$104$	$89$	$C_8^2$	$S_3$	$C_3$	$C_{8} \times C_{16}$	$2^{12} \cdot 3$	$2^{11}$	$384$	$27$	$-$	$-$	$-$	$3$	$6$	$14$	Solvable - 2
384.13	$C_6.C_8^2$	$2^{7} \cdot 3$	$2^{4} \cdot 3$	$-1$	$2$	$8$	$2$	144	$134$	$96$	$73$	$C_4\times C_8$	$D_6$	$C_6$	$C_{8}^{2}$	$2^{12} \cdot 3$	$2^{10}$	$384$	$27$	$-$	$-$	$-$	$3$	$6$	$14$	Solvable - 2
384.14	$C_{24}:C_{16}$	$2^{7} \cdot 3$	$2^{4} \cdot 3$	$-1$	$2$	$8$	$2$	144	$134$	$92$	$73$	$C_4\times C_8$	$D_6$	$C_6$	$C_{4} \times C_{16}$	$2^{11} \cdot 3$	$2^{9}$	$384$	$27$	$-$	$-$	$-$	$3$	$6$	$14$	Solvable - 2
384.15	$C_{24}:C_{16}$	$2^{7} \cdot 3$	$2^{4} \cdot 3$	$-1$	$2$	$8$	$2$	120	$134$	$76$	$53$	$C_2\times C_8$	$D_{12}$	$C_{12}$	$C_{2} \times C_{16}$	$2^{10} \cdot 3$	$2^{7}$	$384$	$27$	$-$	$-$	$-$	$3$	$6$	$14$	Solvable - 2
384.16	$C_{24}:C_{16}$	$2^{7} \cdot 3$	$2^{4} \cdot 3$	$-1$	$2$	$8$	$2$	120	$134$	$76$	$53$	$C_2\times C_8$	$D_{12}$	$C_{12}$	$C_{2} \times C_{16}$	$2^{10} \cdot 3$	$2^{7}$	$384$	$27$	$-$	$-$	$-$	$3$	$4$	$14$	Solvable - 2
384.17	$D_{24}:C_8$	$2^{7} \cdot 3$	$2^{4} \cdot 3$	$-1$	$2$	$8$	$2$	108	$326$	$100$	$37$	$C_8$	$D_{24}$	$C_{24}$	$C_{2} \times C_{8}$	$2^{9} \cdot 3$	$2^{5}$	$48$	$19$	$2$	$4$	$16$	$2$	$4$	$10$	Solvable - 2
384.18	$C_{24}.\OD_{16}$	$2^{7} \cdot 3$	$2^{4} \cdot 3$	$-1$	$2$	$8$	$2$	84	$134$	$80$	$53$	$C_2\times C_4$	$C_{12}:C_4$	$C_{12}$	$C_{4} \times C_{8}$	$2^{10} \cdot 3$	$2^{6}$	$384$	$27$	$-$	$-$	$-$	$5$	$10$	$14$	Solvable - 2
384.19	$D_{24}:C_8$	$2^{7} \cdot 3$	$2^{4} \cdot 3$	$-1$	$2$	$8$	$2$	72	$326$	$96$	$37$	$C_4$	$D_{12}:C_4$	$C_{24}$	$C_{2} \times C_{8}$	$2^{9} \cdot 3$	$2^{4}$	$48$	$19$	$4$	$8$	$16$	$4$	$8$	$10$	Solvable - 2
384.20	$(C_2^2\times D_6):C_8$	$2^{7} \cdot 3$	$2^{3} \cdot 3$	$-1$	$2$	$8$	$2$	48	$1702$	$294$	$37$	$C_2^2$	$C_2^2.D_{12}$	$C_2^2\times C_6$	$C_{2} \times C_{8}$	$2^{10} \cdot 3$	$2^{5}$	$48$	$19$	$-$	$-$	$-$	$6$	$8$	$10$	Solvable - 2
384.21	$C_2^3.(S_3\times C_8)$	$2^{7} \cdot 3$	$2^{3} \cdot 3$	$-1$	$2$	$8$	$2$	48	$614$	$150$	$37$	$C_2^2$	$C_2^2.D_{12}$	$C_2^2\times C_6$	$C_{2} \times C_{8}$	$2^{10} \cdot 3$	$2^{5}$	$96$	$27$	$-$	$-$	$-$	$6$	$12$	$14$	Solvable - 2
384.22	$C_2^3.(S_3\times C_8)$	$2^{7} \cdot 3$	$2^{3} \cdot 3$	$-1$	$2$	$8$	$2$	48	$646$	$158$	$37$	$C_2^2$	$C_2^2.D_{12}$	$C_2^2\times C_6$	$C_{2} \times C_{8}$	$2^{10} \cdot 3$	$2^{5}$	$96$	$19$	$-$	$-$	$-$	$6$	$8$	$10$	Solvable - 2
384.23	$C_2^3.(S_3\times C_8)$	$2^{7} \cdot 3$	$2^{3} \cdot 3$	$-1$	$2$	$8$	$2$	48	$646$	$158$	$37$	$C_2^2$	$C_2^2.D_{12}$	$C_2^2\times C_6$	$C_{2} \times C_{8}$	$2^{10} \cdot 3$	$2^{5}$	$96$	$19$	$-$	$-$	$-$	$6$	$8$	$10$	Solvable - 2
384.24	$C_2^3.D_{24}$	$2^{7} \cdot 3$	$2^{3} \cdot 3$	$-1$	$2$	$8$	$2$	45	$1334$	$222$	$35$	$C_2^2$	$C_2^2.D_{12}$	$C_2^2\times C_{12}$	$C_{2} \times C_{4}$	$2^{10} \cdot 3$	$2^{5}$	$48$	$19$	$-$	$-$	$-$	$6$	$6$	$10$	Solvable - 2
384.25	$C_2^4.D_{12}$	$2^{7} \cdot 3$	$2^{3} \cdot 3$	$-1$	$2$	$8$	$2$	45	$790$	$162$	$35$	$C_2^2$	$C_2^2.D_{12}$	$C_2^2\times C_{12}$	$C_{2} \times C_{4}$	$2^{10} \cdot 3$	$2^{5}$	$96$	$19$	$-$	$-$	$-$	$6$	$8$	$10$	Solvable - 2
384.26	$C_2^3.D_{24}$	$2^{7} \cdot 3$	$2^{3} \cdot 3$	$-1$	$2$	$8$	$2$	45	$694$	$146$	$35$	$C_2^2$	$C_2^2.D_{12}$	$C_2^2\times C_{12}$	$C_{2} \times C_{4}$	$2^{10} \cdot 3$	$2^{5}$	$96$	$27$	$-$	$-$	$-$	$6$	$10$	$14$	Solvable - 2
384.27	$C_2^4.D_{12}$	$2^{7} \cdot 3$	$2^{3} \cdot 3$	$-1$	$2$	$8$	$2$	45	$726$	$154$	$35$	$C_2^2$	$C_2^2.D_{12}$	$C_2^2\times C_{12}$	$C_{2} \times C_{4}$	$2^{10} \cdot 3$	$2^{5}$	$96$	$19$	$-$	$-$	$-$	$6$	$8$	$10$	Solvable - 2
384.28	$(C_2^2\times C_4).D_{12}$	$2^{7} \cdot 3$	$2^{3} \cdot 3$	$-1$	$2$	$8$	$2$	48	$438$	$118$	$37$	$C_2^2$	$C_2^2.D_{12}$	$C_2\times C_{12}$	$C_{2} \times C_{8}$	$2^{10} \cdot 3$	$2^{5}$	$96$	$27$	$-$	$-$	$-$	$6$	$10$	$14$	Solvable - 2
384.29	$(C_2^2\times C_4).D_{12}$	$2^{7} \cdot 3$	$2^{3} \cdot 3$	$-1$	$2$	$8$	$2$	48	$502$	$134$	$37$	$C_2^2$	$C_2^2.D_{12}$	$C_2\times C_{12}$	$C_{2} \times C_{8}$	$2^{10} \cdot 3$	$2^{5}$	$96$	$19$	$-$	$-$	$-$	$6$	$8$	$10$	Solvable - 2
384.30	$(C_2\times D_{12}):C_8$	$2^{7} \cdot 3$	$2^{3} \cdot 3$	$-1$	$2$	$8$	$2$	48	$918$	$170$	$37$	$C_2^2$	$C_2^2.D_{12}$	$C_2\times C_{12}$	$C_{2} \times C_{8}$	$2^{10} \cdot 3$	$2^{5}$	$96$	$19$	$-$	$-$	$-$	$6$	$8$	$10$	Solvable - 2
384.31	$(C_2^2\times C_4).D_{12}$	$2^{7} \cdot 3$	$2^{3} \cdot 3$	$-1$	$2$	$8$	$2$	48	$534$	$134$	$37$	$C_2^2$	$C_2^2.D_{12}$	$C_2\times C_{12}$	$C_{2} \times C_{8}$	$2^{10} \cdot 3$	$2^{5}$	$96$	$27$	$-$	$-$	$-$	$6$	$12$	$14$	Solvable - 2
384.32	$(C_2\times C_4).D_{24}$	$2^{7} \cdot 3$	$2^{3} \cdot 3$	$-1$	$2$	$8$	$2$	45	$934$	$168$	$35$	$C_2^2$	$C_2^2.D_{12}$	$C_2^2\times C_{12}$	$C_{2} \times C_{4}$	$2^{10} \cdot 3$	$2^{5}$	$96$	$19$	$-$	$-$	$-$	$6$	$6$	$10$	Solvable - 2
384.33	$(C_2\times C_{12}).Q_{16}$	$2^{7} \cdot 3$	$2^{3} \cdot 3$	$-1$	$2$	$8$	$2$	45	$582$	$140$	$35$	$C_2^2$	$C_2^2.D_{12}$	$C_2^2\times C_{12}$	$C_{2} \times C_{4}$	$2^{10} \cdot 3$	$2^{5}$	$96$	$19$	$-$	$-$	$-$	$6$	$8$	$10$	Solvable - 2
384.34	$(C_2\times C_4).D_{24}$	$2^{7} \cdot 3$	$2^{3} \cdot 3$	$-1$	$2$	$8$	$2$	45	$486$	$124$	$35$	$C_2^2$	$C_2^2.D_{12}$	$C_2^2\times C_{12}$	$C_{2} \times C_{4}$	$2^{10} \cdot 3$	$2^{5}$	$96$	$27$	$-$	$-$	$-$	$6$	$10$	$14$	Solvable - 2
384.35	$(C_2\times C_{12}).Q_{16}$	$2^{7} \cdot 3$	$2^{3} \cdot 3$	$-1$	$2$	$8$	$2$	45	$454$	$116$	$35$	$C_2^2$	$C_2^2.D_{12}$	$C_2^2\times C_{12}$	$C_{2} \times C_{4}$	$2^{10} \cdot 3$	$2^{5}$	$96$	$27$	$-$	$-$	$-$	$6$	$12$	$14$	Solvable - 2
384.36	$(C_4\times C_{12}).D_4$	$2^{7} \cdot 3$	$2^{3} \cdot 3$	$-1$	$3$	$8$	$2$	36	$590$	$112$	$23$	$C_2$	$C_2^3.D_{12}$	$C_6\times Q_8$	$C_{2} \times C_{4}$	$2^{9} \cdot 3$	$2^{3}$	$48$	$19$	$4$	$8$	$16$	$4$	$8$	$10$	Solvable - 3
384.37	$(C_4\times C_{12}).D_4$	$2^{7} \cdot 3$	$2^{3} \cdot 3$	$-1$	$3$	$8$	$2$	36	$590$	$112$	$23$	$C_2$	$C_2^3.D_{12}$	$C_6\times D_4$	$C_{2} \times C_{4}$	$2^{9} \cdot 3$	$2^{3}$	$48$	$19$	$4$	$8$	$16$	$4$	$8$	$10$	Solvable - 3
384.38	$(C_4\times C_{12}).D_4$	$2^{7} \cdot 3$	$2^{3} \cdot 3$	$-1$	$3$	$8$	$2$	36	$718$	$122$	$23$	$C_2$	$C_2^3.D_{12}$	$C_6\times D_4$	$C_{2} \times C_{4}$	$2^{9} \cdot 3$	$2^{3}$	$48$	$19$	$4$	$8$	$16$	$4$	$6$	$10$	Solvable - 3
384.39	$(C_4\times C_{12}).D_4$	$2^{7} \cdot 3$	$2^{3} \cdot 3$	$-1$	$3$	$8$	$2$	36	$494$	$106$	$23$	$C_2$	$C_2^3.D_{12}$	$C_6\times Q_8$	$C_{2} \times C_{4}$	$2^{9} \cdot 3$	$2^{3}$	$48$	$19$	$4$	$8$	$16$	$4$	$8$	$10$	Solvable - 3
384.40	$C_{24}.\OD_{16}$	$2^{7} \cdot 3$	$2^{4} \cdot 3$	$-1$	$2$	$8$	$2$	60	$166$	$68$	$41$	$C_2^2$	$C_{12}.Q_8$	$C_{24}$	$C_{2} \times C_{8}$	$2^{9} \cdot 3$	$2^{4}$	$384$	$27$	$-$	$-$	$-$	$5$	$10$	$14$	Solvable - 2
384.41	$D_{24}:C_8$	$2^{7} \cdot 3$	$2^{4} \cdot 3$	$-1$	$2$	$8$	$2$	60	$518$	$108$	$41$	$C_2^2$	$D_{12}:C_4$	$C_{24}$	$C_{2} \times C_{8}$	$2^{10} \cdot 3$	$2^{5}$	$192$	$27$	$-$	$-$	$-$	$5$	$6$	$14$	Solvable - 2
384.42	$C_{24}.\OD_{16}$	$2^{7} \cdot 3$	$2^{4} \cdot 3$	$-1$	$2$	$8$	$2$	60	$326$	$96$	$41$	$C_2^2$	$D_{12}:C_4$	$C_{24}$	$C_{2} \times C_{8}$	$2^{10} \cdot 3$	$2^{5}$	$384$	$27$	$-$	$-$	$-$	$5$	$10$	$14$	Solvable - 2
384.43	$C_8.D_{24}$	$2^{7} \cdot 3$	$2^{4} \cdot 3$	$-1$	$2$	$8$	$2$	57	$534$	$102$	$39$	$C_2^2$	$D_{12}:C_4$	$C_2\times C_{24}$	$C_{2} \times C_{4}$	$2^{10} \cdot 3$	$2^{5}$	$192$	$27$	$-$	$-$	$-$	$4$	$6$	$14$	Solvable - 2
384.44	$C_{24}.Q_{16}$	$2^{7} \cdot 3$	$2^{4} \cdot 3$	$-1$	$2$	$8$	$2$	57	$278$	$80$	$39$	$C_2^2$	$D_{12}:C_4$	$C_2\times C_{24}$	$C_{2} \times C_{4}$	$2^{10} \cdot 3$	$2^{5}$	$384$	$27$	$-$	$-$	$-$	$4$	$8$	$14$	Solvable - 2
384.45	$C_{12}.D_{16}$	$2^{7} \cdot 3$	$2^{4} \cdot 3$	$-1$	$2$	$8$	$2$	72	$166$	$70$	$45$	$C_2\times C_4$	$C_3:D_8$	$C_{24}$	$C_{2} \times C_{8}$	$2^{10} \cdot 3$	$2^{6}$	$384$	$27$	$-$	$-$	$-$	$4$	$6$	$14$	Solvable - 2
384.46	$C_{12}.\SD_{32}$	$2^{7} \cdot 3$	$2^{4} \cdot 3$	$-1$	$2$	$8$	$2$	72	$166$	$70$	$45$	$C_2\times C_4$	$C_3:D_8$	$C_{24}$	$C_{2} \times C_{8}$	$2^{10} \cdot 3$	$2^{6}$	$384$	$27$	$-$	$-$	$-$	$4$	$6$	$14$	Solvable - 2
384.47	$D_{24}:C_8$	$2^{7} \cdot 3$	$2^{4} \cdot 3$	$-1$	$2$	$8$	$2$	72	$518$	$110$	$45$	$C_2\times C_4$	$C_3:D_8$	$C_{24}$	$C_{2} \times C_{8}$	$2^{10} \cdot 3$	$2^{6}$	$192$	$27$	$-$	$-$	$-$	$4$	$6$	$14$	Solvable - 2
384.48	$C_{12}.Q_{32}$	$2^{7} \cdot 3$	$2^{4} \cdot 3$	$-1$	$2$	$8$	$2$	72	$326$	$98$	$45$	$C_2\times C_4$	$C_3:D_8$	$C_{24}$	$C_{2} \times C_{8}$	$2^{10} \cdot 3$	$2^{6}$	$384$	$27$	$-$	$-$	$-$	$4$	$8$	$14$	Solvable - 2
384.49	$C_8.D_{24}$	$2^{7} \cdot 3$	$2^{4} \cdot 3$	$-1$	$2$	$8$	$2$	57	$790$	$118$	$43$	$C_2^2$	$D_{12}:C_4$	$C_2\times C_{24}$	$C_{2} \times C_{4}$	$2^{11} \cdot 3$	$2^{6}$	$192$	$27$	$-$	$-$	$-$	$4$	$4$	$14$	Solvable - 2
384.50	$C_8.D_{24}$	$2^{7} \cdot 3$	$2^{4} \cdot 3$	$-1$	$2$	$8$	$2$	57	$406$	$98$	$43$	$C_2^2$	$D_{12}:C_4$	$C_2\times C_{24}$	$C_{2} \times C_{4}$	$2^{11} \cdot 3$	$2^{6}$	$384$	$27$	$-$	$-$	$-$	$4$	$6$	$14$	Solvable - 2

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Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$ over number fields
Genus 2 curves
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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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.

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Learn more

Refine search Advanced search options

Results (1-50 of 20169 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