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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 3548 matches)

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Label	Name	Family 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
1.1	$C_1$	$C_{1}$	$1$	$1$	$0$	$0$	$0$	$0$	1	$1$	$1$	$1$	$C_1$	$C_1$	$C_1$	$C_1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$0$	$0$	$0$	Trivial
2.1	$C_2$	$C_{2}$	$2$	$2$	$1$	$1$	$1$	$1$	2	$2$	$2$	$2$	$C_2$	$C_1$	$C_1$	$C_{2}$	$1$	$1$	$2$	$2$	$1$	$1$	$1$	$1$	$1$	$1$	Cyclic
3.1	$C_3$	$C_{3}$	$3$	$3$	$1$	$1$	$1$	$1$	3	$2$	$2$	$2$	$C_3$	$C_1$	$C_1$	$C_{3}$	$2$	$2$	$3$	$3$	$1$	$2$	$2$	$1$	$2$	$2$	Cyclic
4.1	$C_4$	$C_{4}$	$2^{2}$	$2^{2}$	$1$	$1$	$2$	$1$	4	$3$	$3$	$3$	$C_4$	$C_1$	$C_1$	$C_{4}$	$2$	$2$	$4$	$4$	$1$	$2$	$2$	$1$	$2$	$2$	Cyclic
5.1	$C_5$	$C_{5}$	$5$	$5$	$1$	$1$	$1$	$1$	5	$2$	$2$	$2$	$C_5$	$C_1$	$C_1$	$C_{5}$	$2^{2}$	$2^{2}$	$5$	$5$	$1$	$2$	$4$	$1$	$2$	$4$	Cyclic
6.2	$C_6$	$C_{6}$	$2 \cdot 3$	$2 \cdot 3$	$1$	$1$	$2$	$1$	6	$4$	$4$	$4$	$C_6$	$C_1$	$C_1$	$C_{6}$	$2$	$2$	$6$	$5$	$1$	$2$	$2$	$1$	$2$	$2$	Cyclic
7.1	$C_7$	$C_{7}$	$7$	$7$	$1$	$1$	$1$	$1$	7	$2$	$2$	$2$	$C_7$	$C_1$	$C_1$	$C_{7}$	$2 \cdot 3$	$2 \cdot 3$	$7$	$7$	$1$	$2$	$6$	$1$	$2$	$6$	Cyclic
8.1	$C_8$	$C_{8}$	$2^{3}$	$2^{3}$	$1$	$1$	$3$	$1$	8	$4$	$4$	$4$	$C_8$	$C_1$	$C_1$	$C_{8}$	$2^{2}$	$2^{2}$	$8$	$8$	$1$	$2$	$4$	$1$	$2$	$4$	Cyclic
9.1	$C_9$	$C_{9}$	$3^{2}$	$3^{2}$	$1$	$1$	$2$	$1$	9	$3$	$3$	$3$	$C_9$	$C_1$	$C_1$	$C_{9}$	$2 \cdot 3$	$2 \cdot 3$	$9$	$9$	$1$	$2$	$6$	$1$	$2$	$6$	Cyclic
10.2	$C_{10}$	$C_{10}$	$2 \cdot 5$	$2 \cdot 5$	$1$	$1$	$2$	$1$	10	$4$	$4$	$4$	$C_{10}$	$C_1$	$C_1$	$C_{10}$	$2^{2}$	$2^{2}$	$10$	$7$	$1$	$2$	$4$	$1$	$2$	$4$	Cyclic
11.1	$C_{11}$	$C_{11}$	$11$	$11$	$1$	$1$	$1$	$1$	11	$2$	$2$	$2$	$C_{11}$	$C_1$	$C_1$	$C_{11}$	$2 \cdot 5$	$2 \cdot 5$	$11$	$11$	$1$	$2$	$10$	$1$	$2$	$10$	Cyclic
12.2	$C_{12}$	$C_{12}$	$2^{2} \cdot 3$	$2^{2} \cdot 3$	$1$	$1$	$3$	$1$	12	$6$	$6$	$6$	$C_{12}$	$C_1$	$C_1$	$C_{12}$	$2^{2}$	$2^{2}$	$12$	$7$	$1$	$2$	$4$	$1$	$2$	$4$	Cyclic
13.1	$C_{13}$	$C_{13}$	$13$	$13$	$1$	$1$	$1$	$1$	13	$2$	$2$	$2$	$C_{13}$	$C_1$	$C_1$	$C_{13}$	$2^{2} \cdot 3$	$2^{2} \cdot 3$	$13$	$13$	$1$	$2$	$12$	$1$	$2$	$12$	Cyclic
14.2	$C_{14}$	$C_{14}$	$2 \cdot 7$	$2 \cdot 7$	$1$	$1$	$2$	$1$	14	$4$	$4$	$4$	$C_{14}$	$C_1$	$C_1$	$C_{14}$	$2 \cdot 3$	$2 \cdot 3$	$14$	$9$	$1$	$2$	$6$	$1$	$2$	$6$	Cyclic
15.1	$C_{15}$	$C_{15}$	$3 \cdot 5$	$3 \cdot 5$	$1$	$1$	$2$	$1$	15	$4$	$4$	$4$	$C_{15}$	$C_1$	$C_1$	$C_{15}$	$2^{3}$	$2^{3}$	$15$	$8$	$1$	$2$	$8$	$1$	$2$	$6$	Cyclic
16.1	$C_{16}$	$C_{16}$	$2^{4}$	$2^{4}$	$1$	$1$	$4$	$1$	16	$5$	$5$	$5$	$C_{16}$	$C_1$	$C_1$	$C_{16}$	$2^{3}$	$2^{3}$	$16$	$16$	$1$	$2$	$8$	$1$	$2$	$8$	Cyclic
17.1	$C_{17}$	$C_{17}$	$17$	$17$	$1$	$1$	$1$	$1$	17	$2$	$2$	$2$	$C_{17}$	$C_1$	$C_1$	$C_{17}$	$2^{4}$	$2^{4}$	$17$	$17$	$1$	$2$	$16$	$1$	$2$	$16$	Cyclic
18.2	$C_{18}$	$C_{18}$	$2 \cdot 3^{2}$	$2 \cdot 3^{2}$	$1$	$1$	$3$	$1$	18	$6$	$6$	$6$	$C_{18}$	$C_1$	$C_1$	$C_{18}$	$2 \cdot 3$	$2 \cdot 3$	$18$	$11$	$1$	$2$	$6$	$1$	$2$	$6$	Cyclic
19.1	$C_{19}$	$C_{19}$	$19$	$19$	$1$	$1$	$1$	$1$	19	$2$	$2$	$2$	$C_{19}$	$C_1$	$C_1$	$C_{19}$	$2 \cdot 3^{2}$	$2 \cdot 3^{2}$	$19$	$19$	$1$	$2$	$18$	$1$	$2$	$18$	Cyclic
20.2	$C_{20}$	$C_{20}$	$2^{2} \cdot 5$	$2^{2} \cdot 5$	$1$	$1$	$3$	$1$	20	$6$	$6$	$6$	$C_{20}$	$C_1$	$C_1$	$C_{20}$	$2^{3}$	$2^{3}$	$20$	$9$	$1$	$2$	$8$	$1$	$2$	$6$	Cyclic
21.2	$C_{21}$	$C_{21}$	$3 \cdot 7$	$3 \cdot 7$	$1$	$1$	$2$	$1$	21	$4$	$4$	$4$	$C_{21}$	$C_1$	$C_1$	$C_{21}$	$2^{2} \cdot 3$	$2^{2} \cdot 3$	$21$	$10$	$1$	$2$	$12$	$1$	$2$	$8$	Cyclic
22.2	$C_{22}$	$C_{22}$	$2 \cdot 11$	$2 \cdot 11$	$1$	$1$	$2$	$1$	22	$4$	$4$	$4$	$C_{22}$	$C_1$	$C_1$	$C_{22}$	$2 \cdot 5$	$2 \cdot 5$	$22$	$13$	$1$	$2$	$10$	$1$	$2$	$10$	Cyclic
23.1	$C_{23}$	$C_{23}$	$23$	$23$	$1$	$1$	$1$	$1$	23	$2$	$2$	$2$	$C_{23}$	$C_1$	$C_1$	$C_{23}$	$2 \cdot 11$	$2 \cdot 11$	$23$	$23$	$1$	$2$	$22$	$1$	$2$	$22$	Cyclic
24.2	$C_{24}$	$C_{24}$	$2^{3} \cdot 3$	$2^{3} \cdot 3$	$1$	$1$	$4$	$1$	24	$8$	$8$	$8$	$C_{24}$	$C_1$	$C_1$	$C_{24}$	$2^{3}$	$2^{3}$	$24$	$11$	$1$	$2$	$8$	$1$	$2$	$6$	Cyclic
25.1	$C_{25}$	$C_{25}$	$5^{2}$	$5^{2}$	$1$	$1$	$2$	$1$	25	$3$	$3$	$3$	$C_{25}$	$C_1$	$C_1$	$C_{25}$	$2^{2} \cdot 5$	$2^{2} \cdot 5$	$25$	$25$	$1$	$2$	$20$	$1$	$2$	$20$	Cyclic
26.2	$C_{26}$	$C_{26}$	$2 \cdot 13$	$2 \cdot 13$	$1$	$1$	$2$	$1$	26	$4$	$4$	$4$	$C_{26}$	$C_1$	$C_1$	$C_{26}$	$2^{2} \cdot 3$	$2^{2} \cdot 3$	$26$	$15$	$1$	$2$	$12$	$1$	$2$	$12$	Cyclic
27.1	$C_{27}$	$C_{27}$	$3^{3}$	$3^{3}$	$1$	$1$	$3$	$1$	27	$4$	$4$	$4$	$C_{27}$	$C_1$	$C_1$	$C_{27}$	$2 \cdot 3^{2}$	$2 \cdot 3^{2}$	$27$	$27$	$1$	$2$	$18$	$1$	$2$	$18$	Cyclic
28.2	$C_{28}$	$C_{28}$	$2^{2} \cdot 7$	$2^{2} \cdot 7$	$1$	$1$	$3$	$1$	28	$6$	$6$	$6$	$C_{28}$	$C_1$	$C_1$	$C_{28}$	$2^{2} \cdot 3$	$2^{2} \cdot 3$	$28$	$11$	$1$	$2$	$12$	$1$	$2$	$8$	Cyclic
29.1	$C_{29}$	$C_{29}$	$29$	$29$	$1$	$1$	$1$	$1$	29	$2$	$2$	$2$	$C_{29}$	$C_1$	$C_1$	$C_{29}$	$2^{2} \cdot 7$	$2^{2} \cdot 7$	$29$	$29$	$1$	$2$	$28$	$1$	$2$	$28$	Cyclic
30.4	$C_{30}$	$C_{30}$	$2 \cdot 3 \cdot 5$	$2 \cdot 3 \cdot 5$	$1$	$1$	$3$	$1$	30	$8$	$8$	$8$	$C_{30}$	$C_1$	$C_1$	$C_{30}$	$2^{3}$	$2^{3}$	$30$	$10$	$1$	$2$	$8$	$1$	$2$	$6$	Cyclic
31.1	$C_{31}$	$C_{31}$	$31$	$31$	$1$	$1$	$1$	$1$	31	$2$	$2$	$2$	$C_{31}$	$C_1$	$C_1$	$C_{31}$	$2 \cdot 3 \cdot 5$	$2 \cdot 3 \cdot 5$	$31$	$31$	$1$	$2$	$30$	$1$	$2$	$30$	Cyclic
32.1	$C_{32}$	$C_{32}$	$2^{5}$	$2^{5}$	$1$	$1$	$5$	$1$	32	$6$	$6$	$6$	$C_{32}$	$C_1$	$C_1$	$C_{32}$	$2^{4}$	$2^{4}$	$32$	$32$	$1$	$2$	$16$	$1$	$2$	$16$	Cyclic
33.1	$C_{33}$	$C_{33}$	$3 \cdot 11$	$3 \cdot 11$	$1$	$1$	$2$	$1$	33	$4$	$4$	$4$	$C_{33}$	$C_1$	$C_1$	$C_{33}$	$2^{2} \cdot 5$	$2^{2} \cdot 5$	$33$	$14$	$1$	$2$	$20$	$1$	$2$	$12$	Cyclic
34.2	$C_{34}$	$C_{34}$	$2 \cdot 17$	$2 \cdot 17$	$1$	$1$	$2$	$1$	34	$4$	$4$	$4$	$C_{34}$	$C_1$	$C_1$	$C_{34}$	$2^{4}$	$2^{4}$	$34$	$19$	$1$	$2$	$16$	$1$	$2$	$16$	Cyclic
35.1	$C_{35}$	$C_{35}$	$5 \cdot 7$	$5 \cdot 7$	$1$	$1$	$2$	$1$	35	$4$	$4$	$4$	$C_{35}$	$C_1$	$C_1$	$C_{35}$	$2^{3} \cdot 3$	$2^{3} \cdot 3$	$35$	$12$	$1$	$2$	$24$	$1$	$2$	$10$	Cyclic
36.2	$C_{36}$	$C_{36}$	$2^{2} \cdot 3^{2}$	$2^{2} \cdot 3^{2}$	$1$	$1$	$4$	$1$	36	$9$	$9$	$9$	$C_{36}$	$C_1$	$C_1$	$C_{36}$	$2^{2} \cdot 3$	$2^{2} \cdot 3$	$36$	$13$	$1$	$2$	$12$	$1$	$2$	$8$	Cyclic
37.1	$C_{37}$	$C_{37}$	$37$	$37$	$1$	$1$	$1$	$1$	37	$2$	$2$	$2$	$C_{37}$	$C_1$	$C_1$	$C_{37}$	$2^{2} \cdot 3^{2}$	$2^{2} \cdot 3^{2}$	$37$	$37$	$1$	$2$	$36$	$1$	$2$	$36$	Cyclic
38.2	$C_{38}$	$C_{38}$	$2 \cdot 19$	$2 \cdot 19$	$1$	$1$	$2$	$1$	38	$4$	$4$	$4$	$C_{38}$	$C_1$	$C_1$	$C_{38}$	$2 \cdot 3^{2}$	$2 \cdot 3^{2}$	$38$	$21$	$1$	$2$	$18$	$1$	$2$	$18$	Cyclic
39.2	$C_{39}$	$C_{39}$	$3 \cdot 13$	$3 \cdot 13$	$1$	$1$	$2$	$1$	39	$4$	$4$	$4$	$C_{39}$	$C_1$	$C_1$	$C_{39}$	$2^{3} \cdot 3$	$2^{3} \cdot 3$	$39$	$16$	$1$	$2$	$24$	$1$	$2$	$14$	Cyclic
40.2	$C_{40}$	$C_{40}$	$2^{3} \cdot 5$	$2^{3} \cdot 5$	$1$	$1$	$4$	$1$	40	$8$	$8$	$8$	$C_{40}$	$C_1$	$C_1$	$C_{40}$	$2^{4}$	$2^{4}$	$40$	$13$	$1$	$2$	$16$	$1$	$2$	$8$	Cyclic
41.1	$C_{41}$	$C_{41}$	$41$	$41$	$1$	$1$	$1$	$1$	41	$2$	$2$	$2$	$C_{41}$	$C_1$	$C_1$	$C_{41}$	$2^{3} \cdot 5$	$2^{3} \cdot 5$	$41$	$41$	$1$	$2$	$40$	$1$	$2$	$40$	Cyclic
42.6	$C_{42}$	$C_{42}$	$2 \cdot 3 \cdot 7$	$2 \cdot 3 \cdot 7$	$1$	$1$	$3$	$1$	42	$8$	$8$	$8$	$C_{42}$	$C_1$	$C_1$	$C_{42}$	$2^{2} \cdot 3$	$2^{2} \cdot 3$	$42$	$12$	$1$	$2$	$12$	$1$	$2$	$8$	Cyclic
43.1	$C_{43}$	$C_{43}$	$43$	$43$	$1$	$1$	$1$	$1$	43	$2$	$2$	$2$	$C_{43}$	$C_1$	$C_1$	$C_{43}$	$2 \cdot 3 \cdot 7$	$2 \cdot 3 \cdot 7$	$43$	$43$	$1$	$2$	$42$	$1$	$2$	$42$	Cyclic
44.2	$C_{44}$	$C_{44}$	$2^{2} \cdot 11$	$2^{2} \cdot 11$	$1$	$1$	$3$	$1$	44	$6$	$6$	$6$	$C_{44}$	$C_1$	$C_1$	$C_{44}$	$2^{2} \cdot 5$	$2^{2} \cdot 5$	$44$	$15$	$1$	$2$	$20$	$1$	$2$	$12$	Cyclic
45.1	$C_{45}$	$C_{45}$	$3^{2} \cdot 5$	$3^{2} \cdot 5$	$1$	$1$	$3$	$1$	45	$6$	$6$	$6$	$C_{45}$	$C_1$	$C_1$	$C_{45}$	$2^{3} \cdot 3$	$2^{3} \cdot 3$	$45$	$14$	$1$	$2$	$24$	$1$	$2$	$10$	Cyclic
46.2	$C_{46}$	$C_{46}$	$2 \cdot 23$	$2 \cdot 23$	$1$	$1$	$2$	$1$	46	$4$	$4$	$4$	$C_{46}$	$C_1$	$C_1$	$C_{46}$	$2 \cdot 11$	$2 \cdot 11$	$46$	$25$	$1$	$2$	$22$	$1$	$2$	$22$	Cyclic
47.1	$C_{47}$	$C_{47}$	$47$	$47$	$1$	$1$	$1$	$1$	47	$2$	$2$	$2$	$C_{47}$	$C_1$	$C_1$	$C_{47}$	$2 \cdot 23$	$2 \cdot 23$	$47$	$47$	$1$	$2$	$46$	$1$	$2$	$46$	Cyclic
48.2	$C_{48}$	$C_{48}$	$2^{4} \cdot 3$	$2^{4} \cdot 3$	$1$	$1$	$5$	$1$	48	$10$	$10$	$10$	$C_{48}$	$C_1$	$C_1$	$C_{48}$	$2^{4}$	$2^{4}$	$48$	$19$	$1$	$2$	$16$	$1$	$2$	$10$	Cyclic
49.1	$C_{49}$	$C_{49}$	$7^{2}$	$7^{2}$	$1$	$1$	$2$	$1$	49	$3$	$3$	$3$	$C_{49}$	$C_1$	$C_1$	$C_{49}$	$2 \cdot 3 \cdot 7$	$2 \cdot 3 \cdot 7$	$49$	$49$	$1$	$2$	$42$	$1$	$2$	$42$	Cyclic
50.2	$C_{50}$	$C_{50}$	$2 \cdot 5^{2}$	$2 \cdot 5^{2}$	$1$	$1$	$3$	$1$	50	$6$	$6$	$6$	$C_{50}$	$C_1$	$C_1$	$C_{50}$	$2^{2} \cdot 5$	$2^{2} \cdot 5$	$50$	$27$	$1$	$2$	$20$	$1$	$2$	$20$	Cyclic

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