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 3416 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
129.1	$C_{43}:C_3$	$3 \cdot 43$	$3 \cdot 43$	$-1$	$2$	$2$	$2$	17	$46$	$4$	$3$	$C_1$	$C_{43}:C_3$	$C_{43}$	$C_{3}$	$2 \cdot 3 \cdot 7 \cdot 43$	$2 \cdot 7$	$43$	$43$	$3$	$6$	$42$	$3$	$6$	$42$	Solvable - 2
129.2	$C_{129}$	$3 \cdot 43$	$3 \cdot 43$	$1$	$1$	$2$	$1$	129	$4$	$4$	$4$	$C_{129}$	$C_1$	$C_1$	$C_{129}$	$2^{2} \cdot 3 \cdot 7$	$2^{2} \cdot 3 \cdot 7$	$129$	$46$	$1$	$2$	$84$	$1$	$2$	$44$	Cyclic
130.1	$D_5\times C_{13}$	$2 \cdot 5 \cdot 13$	$2 \cdot 5 \cdot 13$	$-1$	$2$	$3$	$2$	52	$16$	$8$	$6$	$C_{13}$	$D_5$	$C_5$	$C_{26}$	$2^{4} \cdot 3 \cdot 5$	$2^{3} \cdot 3$	$65$	$18$	$2$	$4$	$48$	$2$	$4$	$16$	Solvable - 2
130.2	$C_5\times D_{13}$	$2 \cdot 5 \cdot 13$	$2 \cdot 5 \cdot 13$	$-1$	$2$	$3$	$2$	40	$32$	$8$	$6$	$C_5$	$D_{13}$	$C_{13}$	$C_{10}$	$2^{4} \cdot 3 \cdot 13$	$2^{3} \cdot 3$	$65$	$18$	$2$	$4$	$48$	$2$	$4$	$16$	Solvable - 2
130.3	$D_{65}$	$2 \cdot 5 \cdot 13$	$2 \cdot 5 \cdot 13$	$-1$	$2$	$3$	$2$	34	$88$	$8$	$5$	$C_1$	$D_{65}$	$C_{65}$	$C_{2}$	$2^{4} \cdot 3 \cdot 5 \cdot 13$	$2^{3} \cdot 3$	$65$	$18$	$2$	$2$	$48$	$2$	$2$	$16$	Solvable - 2
130.4	$C_{130}$	$2 \cdot 5 \cdot 13$	$2 \cdot 5 \cdot 13$	$1$	$1$	$3$	$1$	130	$8$	$8$	$8$	$C_{130}$	$C_1$	$C_1$	$C_{130}$	$2^{4} \cdot 3$	$2^{4} \cdot 3$	$130$	$20$	$1$	$2$	$48$	$1$	$2$	$16$	Cyclic
131.1	$C_{131}$	$131$	$131$	$1$	$1$	$1$	$1$	131	$2$	$2$	$2$	$C_{131}$	$C_1$	$C_1$	$C_{131}$	$2 \cdot 5 \cdot 13$	$2 \cdot 5 \cdot 13$	$131$	$131$	$1$	$2$	$130$	$1$	$2$	$130$	Cyclic
132.1	$C_3:C_{44}$	$2^{2} \cdot 3 \cdot 11$	$2^{2} \cdot 3 \cdot 11$	$-1$	$2$	$4$	$2$	66	$16$	$12$	$10$	$C_{22}$	$S_3$	$C_3$	$C_{44}$	$2^{3} \cdot 3 \cdot 5$	$2^{2} \cdot 5$	$132$	$18$	$2$	$4$	$40$	$2$	$4$	$14$	Solvable - 2
132.2	$C_{11}:C_{12}$	$2^{2} \cdot 3 \cdot 11$	$2^{2} \cdot 3 \cdot 11$	$-1$	$2$	$4$	$2$	42	$32$	$12$	$10$	$C_6$	$D_{11}$	$C_{11}$	$C_{12}$	$2^{3} \cdot 5 \cdot 11$	$2^{2} \cdot 5$	$132$	$18$	$2$	$4$	$20$	$2$	$4$	$14$	Solvable - 2
132.3	$C_{33}:C_4$	$2^{2} \cdot 3 \cdot 11$	$2^{2} \cdot 3 \cdot 11$	$-1$	$2$	$4$	$2$	36	$56$	$12$	$9$	$C_2$	$D_{33}$	$C_{33}$	$C_{4}$	$2^{3} \cdot 3 \cdot 5 \cdot 11$	$2^{2} \cdot 5$	$132$	$18$	$2$	$4$	$40$	$2$	$4$	$14$	Solvable - 2
132.4	$C_{132}$	$2^{2} \cdot 3 \cdot 11$	$2^{2} \cdot 3 \cdot 11$	$1$	$1$	$4$	$1$	132	$12$	$12$	$12$	$C_{132}$	$C_1$	$C_1$	$C_{132}$	$2^{3} \cdot 5$	$2^{3} \cdot 5$	$132$	$18$	$1$	$2$	$40$	$1$	$2$	$14$	Cyclic
132.5	$S_3\times D_{11}$	$2^{2} \cdot 3 \cdot 11$	$2 \cdot 3 \cdot 11$	$-1$	$2$	$4$	$2$	21	$132$	$20$	$10$	$C_1$	$S_3\times D_{11}$	$C_{33}$	$C_{2}^{2}$	$2^{2} \cdot 3 \cdot 5 \cdot 11$	$5$	$33$	$14$	$4$	$4$	$20$	$4$	$4$	$12$	Solvable - 2
132.6	$C_{11}\times A_4$	$2^{2} \cdot 3 \cdot 11$	$2 \cdot 3 \cdot 11$	$-1$	$2$	$4$	$2$	44	$20$	$10$	$6$	$C_{11}$	$A_4$	$C_2^2$	$C_{33}$	$2^{4} \cdot 3 \cdot 5$	$2^{2} \cdot 5$	$44$	$15$	$3$	$6$	$30$	$3$	$5$	$13$	Solvable - 2
132.7	$C_3\times D_{22}$	$2^{2} \cdot 3 \cdot 11$	$2 \cdot 3 \cdot 11$	$-1$	$2$	$4$	$2$	42	$80$	$20$	$14$	$C_6$	$D_{11}$	$C_{11}$	$C_{2} \times C_{6}$	$2^{3} \cdot 5 \cdot 11$	$2^{2} \cdot 5$	$66$	$16$	$2$	$4$	$20$	$2$	$4$	$12$	Solvable - 2
132.8	$S_3\times C_{22}$	$2^{2} \cdot 3 \cdot 11$	$2 \cdot 3 \cdot 11$	$-1$	$2$	$4$	$2$	66	$32$	$20$	$14$	$C_{22}$	$S_3$	$C_3$	$C_{2} \times C_{22}$	$2^{3} \cdot 3 \cdot 5$	$2^{2} \cdot 5$	$66$	$16$	$2$	$4$	$20$	$2$	$4$	$12$	Solvable - 2
132.9	$D_{66}$	$2^{2} \cdot 3 \cdot 11$	$2 \cdot 3 \cdot 11$	$-1$	$2$	$4$	$2$	36	$152$	$20$	$11$	$C_2$	$D_{33}$	$C_{33}$	$C_{2}^{2}$	$2^{3} \cdot 3 \cdot 5 \cdot 11$	$2^{2} \cdot 5$	$66$	$16$	$2$	$2$	$20$	$2$	$2$	$12$	Solvable - 2
132.10	$C_2\times C_{66}$	$2^{2} \cdot 3 \cdot 11$	$2 \cdot 3 \cdot 11$	$1$	$1$	$4$	$2$	132	$20$	$20$	$20$	$C_2\times C_{66}$	$C_1$	$C_1$	$C_{2} \times C_{66}$	$2^{3} \cdot 3 \cdot 5$	$2^{3} \cdot 3 \cdot 5$	$132$	$18$	$-$	$-$	$-$	$2$	$3$	$12$	Abelian - 2
133.1	$C_{133}$	$7 \cdot 19$	$7 \cdot 19$	$1$	$1$	$2$	$1$	133	$4$	$4$	$4$	$C_{133}$	$C_1$	$C_1$	$C_{133}$	$2^{2} \cdot 3^{3}$	$2^{2} \cdot 3^{3}$	$133$	$26$	$1$	$2$	$108$	$1$	$2$	$24$	Cyclic
134.1	$D_{67}$	$2 \cdot 67$	$2 \cdot 67$	$-1$	$2$	$2$	$2$	35	$70$	$4$	$3$	$C_1$	$D_{67}$	$C_{67}$	$C_{2}$	$2 \cdot 3 \cdot 11 \cdot 67$	$3 \cdot 11$	$67$	$67$	$2$	$2$	$66$	$2$	$2$	$66$	Solvable - 2
134.2	$C_{134}$	$2 \cdot 67$	$2 \cdot 67$	$1$	$1$	$2$	$1$	134	$4$	$4$	$4$	$C_{134}$	$C_1$	$C_1$	$C_{134}$	$2 \cdot 3 \cdot 11$	$2 \cdot 3 \cdot 11$	$134$	$69$	$1$	$2$	$66$	$1$	$2$	$66$	Cyclic
135.1	$C_{135}$	$3^{3} \cdot 5$	$3^{3} \cdot 5$	$1$	$1$	$4$	$1$	135	$8$	$8$	$8$	$C_{135}$	$C_1$	$C_1$	$C_{135}$	$2^{3} \cdot 3^{2}$	$2^{3} \cdot 3^{2}$	$135$	$32$	$1$	$2$	$72$	$1$	$2$	$22$	Cyclic
135.2	$C_3\times C_{45}$	$3^{3} \cdot 5$	$3^{2} \cdot 5$	$1$	$1$	$4$	$2$	135	$20$	$20$	$20$	$C_3\times C_{45}$	$C_1$	$C_1$	$C_{3} \times C_{45}$	$2^{4} \cdot 3^{3}$	$2^{4} \cdot 3^{3}$	$135$	$17$	$-$	$-$	$-$	$2$	$4$	$12$	Abelian - 2
135.3	$C_5\times \He_3$	$3^{3} \cdot 5$	$3 \cdot 5$	$2$	$2$	$4$	$2$	55	$38$	$22$	$14$	$C_{15}$	$C_3^2$	$C_3$	$C_{3} \times C_{15}$	$2^{6} \cdot 3^{3}$	$2^{6} \cdot 3$	$45$	$14$	$3$	$6$	$24$	$3$	$6$	$10$	Nilpotent - 2
135.4	$C_9:C_{15}$	$3^{3} \cdot 5$	$3^{2} \cdot 5$	$2$	$2$	$4$	$2$	55	$20$	$16$	$14$	$C_{15}$	$C_3^2$	$C_3$	$C_{3} \times C_{15}$	$2^{3} \cdot 3^{3}$	$2^{3} \cdot 3$	$45$	$14$	$3$	$6$	$24$	$3$	$6$	$10$	Nilpotent - 2
135.5	$C_3^2\times C_{15}$	$3^{3} \cdot 5$	$3 \cdot 5$	$1$	$1$	$4$	$3$	135	$56$	$56$	$56$	$C_3^2\times C_{15}$	$C_1$	$C_1$	$C_{3}^{2} \times C_{15}$	$2^{7} \cdot 3^{3} \cdot 13$	$2^{7} \cdot 3^{3} \cdot 13$	$135$	$14$	$-$	$-$	$-$	$3$	$6$	$10$	Abelian - 3
136.1	$C_{17}:C_8$	$2^{3} \cdot 17$	$2^{3} \cdot 17$	$-1$	$2$	$4$	$2$	40	$24$	$8$	$7$	$C_4$	$D_{17}$	$C_{17}$	$C_{8}$	$2^{6} \cdot 17$	$2^{5}$	$136$	$25$	$2$	$4$	$32$	$2$	$4$	$20$	Solvable - 2
136.2	$C_{136}$	$2^{3} \cdot 17$	$2^{3} \cdot 17$	$1$	$1$	$4$	$1$	136	$8$	$8$	$8$	$C_{136}$	$C_1$	$C_1$	$C_{136}$	$2^{6}$	$2^{6}$	$136$	$25$	$1$	$2$	$64$	$1$	$2$	$20$	Cyclic
136.3	$C_{17}:C_8$	$2^{3} \cdot 17$	$2^{3} \cdot 17$	$-1$	$2$	$4$	$2$	16	$40$	$8$	$6$	$C_2$	$C_{17}:C_4$	$C_{17}$	$C_{8}$	$2^{5} \cdot 17$	$2^{3}$	$136$	$25$	$4$	$8$	$32$	$4$	$6$	$20$	Solvable - 2
136.4	$C_{17}:Q_8$	$2^{3} \cdot 17$	$2^{2} \cdot 17$	$-1$	$2$	$4$	$2$	37	$60$	$12$	$9$	$C_2$	$D_{34}$	$C_{34}$	$C_{2}^{2}$	$2^{7} \cdot 17$	$2^{5}$	$136$	$25$	$2$	$4$	$64$	$2$	$4$	$20$	Solvable - 2
136.5	$C_4\times D_{17}$	$2^{3} \cdot 17$	$2^{2} \cdot 17$	$-1$	$2$	$4$	$2$	40	$96$	$16$	$11$	$C_4$	$D_{17}$	$C_{17}$	$C_{2} \times C_{4}$	$2^{6} \cdot 17$	$2^{5}$	$68$	$21$	$2$	$4$	$32$	$2$	$4$	$18$	Solvable - 2
136.6	$D_{68}$	$2^{3} \cdot 17$	$2^{2} \cdot 17$	$-1$	$2$	$4$	$2$	37	$132$	$16$	$9$	$C_2$	$D_{34}$	$C_{34}$	$C_{2}^{2}$	$2^{7} \cdot 17$	$2^{5}$	$68$	$21$	$2$	$2$	$32$	$2$	$2$	$18$	Solvable - 2
136.7	$C_{34}:C_4$	$2^{3} \cdot 17$	$2^{2} \cdot 17$	$-1$	$2$	$4$	$2$	40	$64$	$16$	$13$	$C_2^2$	$D_{17}$	$C_{17}$	$C_{2} \times C_{4}$	$2^{7} \cdot 17$	$2^{6}$	$136$	$23$	$-$	$-$	$-$	$3$	$4$	$18$	Solvable - 2
136.8	$C_{17}:D_4$	$2^{3} \cdot 17$	$2^{2} \cdot 17$	$-1$	$2$	$4$	$2$	37	$100$	$16$	$9$	$C_2$	$D_{34}$	$C_{34}$	$C_{2}^{2}$	$2^{6} \cdot 17$	$2^{4}$	$68$	$21$	$2$	$4$	$32$	$2$	$4$	$18$	Solvable - 2
136.9	$C_2\times C_{68}$	$2^{3} \cdot 17$	$2^{2} \cdot 17$	$1$	$1$	$4$	$2$	136	$16$	$16$	$16$	$C_2\times C_{68}$	$C_1$	$C_1$	$C_{2} \times C_{68}$	$2^{7}$	$2^{7}$	$136$	$23$	$-$	$-$	$-$	$2$	$3$	$18$	Abelian - 2
136.10	$D_4\times C_{17}$	$2^{3} \cdot 17$	$2^{2} \cdot 17$	$2$	$2$	$4$	$2$	85	$20$	$16$	$12$	$C_{34}$	$C_2^2$	$C_2$	$C_{2} \times C_{34}$	$2^{7}$	$2^{5}$	$68$	$21$	$2$	$4$	$32$	$2$	$4$	$18$	Nilpotent - 2
136.11	$Q_8\times C_{17}$	$2^{3} \cdot 17$	$2^{2} \cdot 17$	$2$	$2$	$4$	$2$	85	$12$	$12$	$12$	$C_{34}$	$C_2^2$	$C_2$	$C_{2} \times C_{34}$	$2^{7} \cdot 3$	$2^{5} \cdot 3$	$136$	$25$	$2$	$4$	$32$	$2$	$4$	$20$	Nilpotent - 2
136.12	$C_{17}:C_8$	$2^{3} \cdot 17$	$2^{3} \cdot 17$	$-1$	$2$	$4$	$2$	10	$56$	$8$	$5$	$C_1$	$C_{17}:C_8$	$C_{17}$	$C_{8}$	$2^{4} \cdot 17$	$2$	$17$	$17$	$8$	$8$	$16$	$8$	$8$	$16$	Solvable - 2
136.13	$C_{34}:C_4$	$2^{3} \cdot 17$	$2^{2} \cdot 17$	$-1$	$2$	$4$	$2$	16	$112$	$16$	$10$	$C_2$	$C_{17}:C_4$	$C_{17}$	$C_{2} \times C_{4}$	$2^{5} \cdot 17$	$2^{3}$	$34$	$19$	$4$	$4$	$16$	$4$	$4$	$16$	Solvable - 2
136.14	$C_2\times D_{34}$	$2^{3} \cdot 17$	$2 \cdot 17$	$-1$	$2$	$4$	$3$	40	$208$	$32$	$21$	$C_2^2$	$D_{17}$	$C_{17}$	$C_{2}^{3}$	$2^{7} \cdot 3 \cdot 17$	$2^{6} \cdot 3$	$68$	$21$	$-$	$-$	$-$	$3$	$3$	$17$	Solvable - 2
136.15	$C_2^2\times C_{34}$	$2^{3} \cdot 17$	$2 \cdot 17$	$1$	$1$	$4$	$3$	136	$32$	$32$	$32$	$C_2^2\times C_{34}$	$C_1$	$C_1$	$C_{2}^{2} \times C_{34}$	$2^{7} \cdot 3 \cdot 7$	$2^{7} \cdot 3 \cdot 7$	$136$	$23$	$-$	$-$	$-$	$3$	$4$	$18$	Abelian - 3
137.1	$C_{137}$	$137$	$137$	$1$	$1$	$1$	$1$	137	$2$	$2$	$2$	$C_{137}$	$C_1$	$C_1$	$C_{137}$	$2^{3} \cdot 17$	$2^{3} \cdot 17$	$137$	$137$	$1$	$2$	$136$	$1$	$2$	$136$	Cyclic
138.1	$S_3\times C_{23}$	$2 \cdot 3 \cdot 23$	$2 \cdot 3 \cdot 23$	$-1$	$2$	$3$	$2$	69	$12$	$8$	$6$	$C_{23}$	$S_3$	$C_3$	$C_{46}$	$2^{2} \cdot 3 \cdot 11$	$2 \cdot 11$	$69$	$26$	$2$	$4$	$44$	$2$	$4$	$24$	Solvable - 2
138.2	$C_3\times D_{23}$	$2 \cdot 3 \cdot 23$	$2 \cdot 3 \cdot 23$	$-1$	$2$	$3$	$2$	39	$52$	$8$	$6$	$C_3$	$D_{23}$	$C_{23}$	$C_{6}$	$2^{2} \cdot 11 \cdot 23$	$2 \cdot 11$	$69$	$26$	$2$	$4$	$44$	$2$	$4$	$24$	Solvable - 2
138.3	$D_{69}$	$2 \cdot 3 \cdot 23$	$2 \cdot 3 \cdot 23$	$-1$	$2$	$3$	$2$	36	$100$	$8$	$5$	$C_1$	$D_{69}$	$C_{69}$	$C_{2}$	$2^{2} \cdot 3 \cdot 11 \cdot 23$	$2 \cdot 11$	$69$	$26$	$2$	$2$	$44$	$2$	$2$	$24$	Solvable - 2
138.4	$C_{138}$	$2 \cdot 3 \cdot 23$	$2 \cdot 3 \cdot 23$	$1$	$1$	$3$	$1$	138	$8$	$8$	$8$	$C_{138}$	$C_1$	$C_1$	$C_{138}$	$2^{2} \cdot 11$	$2^{2} \cdot 11$	$138$	$28$	$1$	$2$	$44$	$1$	$2$	$24$	Cyclic
139.1	$C_{139}$	$139$	$139$	$1$	$1$	$1$	$1$	139	$2$	$2$	$2$	$C_{139}$	$C_1$	$C_1$	$C_{139}$	$2 \cdot 3 \cdot 23$	$2 \cdot 3 \cdot 23$	$139$	$139$	$1$	$2$	$138$	$1$	$2$	$138$	Cyclic
140.1	$C_5:C_{28}$	$2^{2} \cdot 5 \cdot 7$	$2^{2} \cdot 5 \cdot 7$	$-1$	$2$	$4$	$2$	56	$20$	$12$	$10$	$C_{14}$	$D_5$	$C_5$	$C_{28}$	$2^{4} \cdot 3 \cdot 5$	$2^{3} \cdot 3$	$140$	$16$	$2$	$4$	$24$	$2$	$4$	$12$	Solvable - 2
140.2	$C_7:C_{20}$	$2^{2} \cdot 5 \cdot 7$	$2^{2} \cdot 5 \cdot 7$	$-1$	$2$	$4$	$2$	50	$24$	$12$	$10$	$C_{10}$	$D_7$	$C_7$	$C_{20}$	$2^{4} \cdot 3 \cdot 7$	$2^{3} \cdot 3$	$140$	$16$	$2$	$4$	$24$	$2$	$4$	$12$	Solvable - 2
140.3	$C_{35}:C_4$	$2^{2} \cdot 5 \cdot 7$	$2^{2} \cdot 5 \cdot 7$	$-1$	$2$	$4$	$2$	38	$56$	$12$	$9$	$C_2$	$D_{35}$	$C_{35}$	$C_{4}$	$2^{4} \cdot 3 \cdot 5 \cdot 7$	$2^{3} \cdot 3$	$140$	$16$	$2$	$4$	$48$	$2$	$4$	$12$	Solvable - 2
140.4	$C_{140}$	$2^{2} \cdot 5 \cdot 7$	$2^{2} \cdot 5 \cdot 7$	$1$	$1$	$4$	$1$	140	$12$	$12$	$12$	$C_{140}$	$C_1$	$C_1$	$C_{140}$	$2^{4} \cdot 3$	$2^{4} \cdot 3$	$140$	$16$	$1$	$2$	$48$	$1$	$2$	$12$	Cyclic

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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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