LMFDB - Abstract group search results

Note: Search results may be incomplete due to uncomputed quantities: Automorphism group order (4830 objects)

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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 84 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
32.46	$C_2^2\times D_4$	$2^{5}$	$2^{2}$	$2$	$2$	$5$	$4$	$20$	$158$	$118$	$78$	$C_2^3$	$C_2^2$	$C_2$	$C_{2}^{4}$	$2^{10} \cdot 3$	$2^{8} \cdot 3$	$16$	$8$	$-$	$-$	$-$	$4$	$4$	$4$	Nilpotent - 2
64.62	$C_2^3.D_4$	$2^{6}$	$2^{2}$	$2$	$2$	$6$	$3$	$28$	$177$	$109$	$49$	$C_2^3$	$C_2^3$	$C_2^2$	$C_{2}^{2} \times C_{4}$	$2^{12}$	$2^{9}$	$32$	$16$	$-$	$-$	$-$	$5$	$8$	$8$	Nilpotent - 2
64.63	$C_4^2:C_4$	$2^{6}$	$2^{2}$	$2$	$2$	$6$	$3$	$28$	$113$	$77$	$49$	$C_2^3$	$C_2^3$	$C_2^2$	$C_{2}^{2} \times C_{4}$	$2^{12}$	$2^{9}$	$64$	$16$	$-$	$-$	$-$	$5$	$8$	$8$	Nilpotent - 2
64.64	$C_4^2:C_4$	$2^{6}$	$2^{2}$	$2$	$2$	$6$	$3$	$28$	$105$	$69$	$41$	$C_2^3$	$C_2^3$	$C_2^2$	$C_{2}^{2} \times C_{4}$	$2^{11} \cdot 3$	$2^{8} \cdot 3$	$64$	$20$	$-$	$-$	$-$	$5$	$10$	$10$	Nilpotent - 2
64.73	$C_2^3:D_4$	$2^{6}$	$2^{2}$	$2$	$2$	$6$	$3$	$22$	$317$	$161$	$43$	$C_2^3$	$C_2^3$	$C_2^3$	$C_{2}^{3}$	$2^{10} \cdot 3$	$2^{7} \cdot 3$	$32$	$12$	$-$	$-$	$-$	?	?	?	Nilpotent - 2
64.74	$C_2^3:Q_8$	$2^{6}$	$2^{2}$	$2$	$2$	$6$	$3$	$22$	$181$	$101$	$39$	$C_2^3$	$C_2^3$	$C_2^3$	$C_{2}^{3}$	$2^{10} \cdot 3$	$2^{7} \cdot 3$	$32$	$16$	$-$	$-$	$-$	$6$	$8$	$8$	Nilpotent - 2
64.76	$C_2^3.C_2^3$	$2^{6}$	$2^{2}$	$2$	$2$	$6$	$3$	$22$	$141$	$91$	$43$	$C_2^3$	$C_2^3$	$C_2^3$	$C_{2}^{3}$	$2^{10} \cdot 3$	$2^{7} \cdot 3$	$64$	$16$	$-$	$-$	$-$	$6$	$8$	$8$	Nilpotent - 2
64.80	$C_2^3.Q_8$	$2^{6}$	$2^{2}$	$2$	$2$	$6$	$3$	$22$	$165$	$93$	$39$	$C_2^3$	$C_2^3$	$C_2^3$	$C_{2}^{3}$	$2^{10} \cdot 3$	$2^{7} \cdot 3$	$32$	$16$	$-$	$-$	$-$	$6$	$8$	$8$	Nilpotent - 2
64.174	$C_4:D_8$	$2^{6}$	$2^{3}$	$3$	$2$	$6$	$3$	$22$	$193$	$81$	$33$	$C_2^2$	$C_2\times D_4$	$C_2\times C_4$	$C_{2}^{3}$	$2^{12}$	$2^{8}$	$32$	$12$	$-$	$-$	$-$	$4$	$4$	$6$	Nilpotent - 3
64.175	$C_4:Q_{16}$	$2^{6}$	$2^{3}$	$3$	$2$	$6$	$3$	$22$	$97$	$61$	$33$	$C_2^2$	$C_2\times D_4$	$C_2\times C_4$	$C_{2}^{3}$	$2^{12}$	$2^{8}$	$64$	$20$	$-$	$-$	$-$	$4$	$6$	$10$	Nilpotent - 3
64.195	$C_4^2:C_2^2$	$2^{6}$	$2^{2}$	$2$	$2$	$6$	$4$	$40$	$201$	$165$	$129$	$C_2^2\times C_4$	$C_2^2$	$C_2$	$C_{2}^{3} \times C_{4}$	$2^{13}$	$2^{11}$	$32$	$14$	$-$	$-$	$-$	?	?	?	Nilpotent - 2
64.196	$C_4^2:C_2^2$	$2^{6}$	$2^{2}$	$2$	$2$	$6$	$4$	$40$	$289$	$213$	$137$	$C_2^2\times C_4$	$C_2^2$	$C_2$	$C_{2}^{3} \times C_{4}$	$2^{12}$	$2^{10}$	$32$	$10$	$-$	$-$	$-$	?	?	?	Nilpotent - 2
64.198	$C_4^2.C_2^2$	$2^{6}$	$2^{2}$	$2$	$2$	$6$	$4$	$40$	$185$	$155$	$125$	$C_4^2$	$C_2^2$	$C_2$	$C_{2}^{3} \times C_{4}$	$2^{10} \cdot 3$	$2^{8} \cdot 3$	$32$	$12$	$-$	$-$	$-$	?	?	?	Nilpotent - 2
64.199	$C_4^2:C_2^2$	$2^{6}$	$2^{2}$	$2$	$2$	$6$	$4$	$34$	$257$	$169$	$121$	$C_2^2$	$C_2^4$	$C_2$	$C_{2}^{3} \times C_{4}$	$2^{12}$	$2^{8}$	$16$	$12$	$-$	$-$	$-$	?	?	?	Nilpotent - 2
64.201	$C_4^2.C_2^2$	$2^{6}$	$2^{2}$	$2$	$2$	$6$	$4$	$34$	$185$	$147$	$121$	$C_2^2$	$C_2^4$	$C_2$	$C_{2}^{3} \times C_{4}$	$2^{10} \cdot 3$	$2^{6} \cdot 3$	$32$	$12$	$-$	$-$	$-$	?	?	?	Nilpotent - 2
64.203	$C_2^3:D_4$	$2^{6}$	$2^{2}$	$2$	$2$	$6$	$4$	$28$	$353$	$213$	$97$	$C_2^3$	$C_2^3$	$C_2^2$	$C_{2}^{4}$	$2^{12}$	$2^{9}$	$32$	$10$	$-$	$-$	$-$	?	?	?	Nilpotent - 2
64.204	$C_2^3:Q_8$	$2^{6}$	$2^{2}$	$2$	$2$	$6$	$4$	$28$	$225$	$161$	$97$	$C_2^3$	$C_2^3$	$C_2^2$	$C_{2}^{4}$	$2^{12}$	$2^{9}$	$32$	$14$	$-$	$-$	$-$	?	?	?	Nilpotent - 2
64.205	$C_2^3.D_4$	$2^{6}$	$2^{2}$	$2$	$2$	$6$	$4$	$28$	$265$	$171$	$89$	$C_2^3$	$C_2^3$	$C_2^2$	$C_{2}^{4}$	$2^{12}$	$2^{9}$	$32$	$14$	$-$	$-$	$-$	?	?	?	Nilpotent - 2
64.207	$C_4^2:C_2^2$	$2^{6}$	$2^{2}$	$2$	$2$	$6$	$4$	$28$	$265$	$165$	$89$	$C_2^3$	$C_2^3$	$C_2^2$	$C_{2}^{4}$	$2^{13}$	$2^{10}$	$32$	$14$	$-$	$-$	$-$	?	?	?	Nilpotent - 2
64.208	$C_4^2.C_2^2$	$2^{6}$	$2^{2}$	$2$	$2$	$6$	$4$	$28$	$137$	$113$	$89$	$C_2^3$	$C_2^3$	$C_2^2$	$C_{2}^{4}$	$2^{13}$	$2^{10}$	$64$	$18$	$-$	$-$	$-$	$5$	$9$	$9$	Nilpotent - 2
64.209	$C_4^2:C_2^2$	$2^{6}$	$2^{2}$	$2$	$2$	$6$	$4$	$28$	$177$	$123$	$81$	$C_2^3$	$C_2^3$	$C_2^2$	$C_{2}^{4}$	$2^{11} \cdot 3$	$2^{8} \cdot 3$	$32$	$18$	$-$	$-$	$-$	$5$	$9$	$9$	Nilpotent - 2
64.224	$C_2^3:Q_8$	$2^{6}$	$2^{2}$	$2$	$2$	$6$	$4$	$22$	$193$	$121$	$81$	$C_2^2$	$C_2^4$	$C_2^2$	$C_{2}^{4}$	$2^{11} \cdot 3$	$2^{7} \cdot 3$	$16$	$16$	$-$	$-$	$-$	$6$	$8$	$8$	Nilpotent - 2
64.230	$D_4\times Q_8$	$2^{6}$	$2^{2}$	$2$	$2$	$6$	$4$	$25$	$189$	$140$	$91$	$C_2^2$	$C_2^4$	$C_2^2$	$C_{2}^{4}$	$2^{10} \cdot 3$	$2^{6} \cdot 3$	$32$	$12$	$-$	$-$	$-$	?	?	?	Nilpotent - 2
64.231	$Q_8:D_4$	$2^{6}$	$2^{2}$	$2$	$2$	$6$	$4$	$25$	$253$	$156$	$83$	$C_2^2$	$C_2^4$	$C_2^2$	$C_{2}^{4}$	$2^{10} \cdot 3$	$2^{6} \cdot 3$	$32$	$12$	$-$	$-$	$-$	?	?	?	Nilpotent - 2
64.238	$Q_8:Q_8$	$2^{6}$	$2^{2}$	$2$	$2$	$6$	$4$	$25$	$117$	$100$	$83$	$C_2^2$	$C_2^4$	$C_2^2$	$C_{2}^{4}$	$2^{10} \cdot 3$	$2^{6} \cdot 3$	$64$	$16$	$-$	$-$	$-$	$4$	$8$	$8$	Nilpotent - 2
64.247	$C_2^2\times \OD_{16}$	$2^{6}$	$2^{3}$	$2$	$2$	$6$	$4$	$40$	$169$	$149$	$129$	$C_2^2\times C_4$	$C_2^2$	$C_2$	$C_{2}^{3} \times C_{4}$	$2^{11} \cdot 3$	$2^{9} \cdot 3$	$32$	$12$	$-$	$-$	$-$	?	?	?	Nilpotent - 2
64.251	$C_2^2\times \SD_{16}$	$2^{6}$	$2^{3}$	$3$	$2$	$6$	$4$	$28$	$249$	$149$	$89$	$C_2^3$	$D_4$	$C_4$	$C_{2}^{4}$	$2^{11} \cdot 3$	$2^{8} \cdot 3$	$32$	$12$	$-$	$-$	$-$	?	?	?	Nilpotent - 3
81.13	$C_9:C_3^2$	$3^{4}$	$3^{2}$	$2$	$2$	$4$	$3$	$33$	$50$	$38$	$32$	$C_3^2$	$C_3^2$	$C_3$	$C_{3}^{3}$	$2^{2} \cdot 3^{6}$	$2^{2} \cdot 3^{4}$	$27$	$12$	$-$	$-$	$-$	$4$	$8$	$8$	Nilpotent - 2
243.3	$C_3^3.C_3^2$	$3^{5}$	$3^{2}$	$3$	$2$	$5$	$2$	$35$	$207$	$47$	$12$	$C_3^2$	$\He_3$	$C_3^3$	$C_{3}^{2}$	$2^{3} \cdot 3^{6}$	$2^{3} \cdot 3^{3}$	$81$	$18$	$-$	$-$	$-$	$6$	$12$	$12$	Nilpotent - 3
243.4	$C_3^3.C_3^2$	$3^{5}$	$3^{2}$	$3$	$2$	$5$	$2$	$35$	$162$	$46$	$12$	$C_3^2$	$\He_3$	$C_3^3$	$C_{3}^{2}$	$2^{2} \cdot 3^{6}$	$2^{2} \cdot 3^{3}$	$81$	$18$	$-$	$-$	$-$	$6$	$12$	$12$	Nilpotent - 3
243.6	$C_3^3.C_3^2$	$3^{5}$	$3^{2}$	$3$	$2$	$5$	$2$	$35$	$144$	$36$	$12$	$C_3^2$	$\He_3$	$C_3^3$	$C_{3}^{2}$	$2^{2} \cdot 3^{6}$	$2^{2} \cdot 3^{3}$	$81$	$36$	$-$	$-$	$-$	$6$	$12$	$24$	Nilpotent - 3
243.7	$C_3^3.C_3^2$	$3^{5}$	$3^{2}$	$3$	$2$	$5$	$2$	$35$	$99$	$35$	$12$	$C_3^2$	$\He_3$	$C_3^3$	$C_{3}^{2}$	$2^{2} \cdot 3^{6}$	$2^{2} \cdot 3^{3}$	$81$	$18$	$-$	$-$	$-$	$6$	$12$	$12$	Nilpotent - 3
243.8	$C_3^3.C_3^2$	$3^{5}$	$3^{2}$	$3$	$2$	$5$	$2$	$35$	$81$	$25$	$12$	$C_3^2$	$\He_3$	$C_3^3$	$C_{3}^{2}$	$2^{2} \cdot 3^{6}$	$2^{2} \cdot 3^{3}$	$81$	$54$	$-$	$-$	$-$	$6$	$12$	$36$	Nilpotent - 3
243.11	$C_{27}:C_9$	$3^{5}$	$3^{3}$	$2$	$2$	$5$	$2$	$99$	$36$	$30$	$27$	$C_3\times C_9$	$C_3^2$	$C_3$	$C_{9}^{2}$	$2 \cdot 3^{7}$	$2 \cdot 3^{5}$	$243$	$36$	$-$	$-$	$-$	$4$	$8$	$24$	Nilpotent - 2
243.12	$C_3^2:C_{27}$	$3^{5}$	$3^{3}$	$2$	$2$	$5$	$2$	$99$	$63$	$35$	$21$	$C_3\times C_9$	$C_3^2$	$C_3$	$C_{3} \times C_{27}$	$2^{2} \cdot 3^{6}$	$2^{2} \cdot 3^{4}$	$81$	$36$	$-$	$-$	$-$	$4$	$8$	$24$	Nilpotent - 2
243.17	$\He_3:C_9$	$3^{5}$	$3^{2}$	$3$	$2$	$5$	$2$	$51$	$153$	$47$	$18$	$C_3^2$	$\He_3$	$C_3^2$	$C_{3} \times C_{9}$	$2^{2} \cdot 3^{6}$	$2^{2} \cdot 3^{3}$	$81$	$18$	$-$	$-$	$-$	$4$	$8$	$12$	Nilpotent - 3
243.27	$C_9^2.C_3$	$3^{5}$	$3^{2}$	$4$	$2$	$5$	$2$	$35$	$63$	$21$	$9$	$C_3$	$\He_3:C_3$	$C_3\times C_9$	$C_{3}^{2}$	$2 \cdot 3^{7}$	$2 \cdot 3^{3}$	$27$	$27$	$3$	$6$	$18$	$3$	$6$	$18$	Nilpotent - 4
243.41	$C_3^2.C_3^3$	$3^{5}$	$3^{2}$	$2$	$2$	$5$	$3$	$51$	$108$	$62$	$39$	$C_3^2$	$C_3^3$	$C_3^2$	$C_{3}^{3}$	$2 \cdot 3^{7}$	$2 \cdot 3^{4}$	$81$	$18$	$-$	$-$	$-$	$6$	$12$	$12$	Nilpotent - 2
243.43	$C_9^2:C_3$	$3^{5}$	$3^{2}$	$2$	$2$	$5$	$3$	$51$	$108$	$60$	$39$	$C_3^2$	$C_3^3$	$C_3^2$	$C_{3}^{3}$	$2^{3} \cdot 3^{6}$	$2^{3} \cdot 3^{3}$	$81$	$18$	$-$	$-$	$-$	$6$	$12$	$12$	Nilpotent - 2
243.44	$C_9^2:C_3$	$3^{5}$	$3^{2}$	$2$	$2$	$5$	$3$	$51$	$90$	$50$	$33$	$C_3^2$	$C_3^3$	$C_3^2$	$C_{3}^{3}$	$2^{3} \cdot 3^{6}$	$2^{3} \cdot 3^{3}$	$81$	$54$	$-$	$-$	$-$	$6$	$12$	$36$	Nilpotent - 2
243.46	$C_9^2:C_3$	$3^{5}$	$3^{2}$	$2$	$2$	$5$	$3$	$51$	$99$	$55$	$36$	$C_3^2$	$C_3^3$	$C_3^2$	$C_{3}^{3}$	$2 \cdot 3^{7}$	$2 \cdot 3^{4}$	$81$	$36$	$-$	$-$	$-$	$6$	$12$	$24$	Nilpotent - 2
243.50	$C_9.C_3^3$	$3^{5}$	$3^{3}$	$2$	$2$	$5$	$3$	$99$	$63$	$55$	$51$	$C_{27}$	$C_3^2$	$C_3$	$C_{3}^{2} \times C_{9}$	$2^{4} \cdot 3^{5}$	$2^{4} \cdot 3^{3}$	$81$	$81$	$3$	$6$	$54$	$3$	$6$	$54$	Nilpotent - 2
243.55	$C_9.\He_3$	$3^{5}$	$3^{2}$	$3$	$2$	$5$	$3$	$51$	$153$	$67$	$33$	$C_9$	$\He_3$	$C_3^2$	$C_{3}^{3}$	$2^{2} \cdot 3^{6}$	$2^{2} \cdot 3^{3}$	$27$	$27$	$3$	$6$	$18$	$3$	$6$	$18$	Nilpotent - 3
343.4	$C_{49}:C_7$	$7^{3}$	$7^{2}$	$2$	$2$	$3$	$2$	$55$	$18$	$12$	$11$	$C_7$	$C_7^2$	$C_7$	$C_{7}^{2}$	$2 \cdot 3 \cdot 7^{3}$	$2 \cdot 3 \cdot 7$	$49$	$49$	$7$	$14$	$42$	$7$	$14$	$42$	Nilpotent - 2
625.6	$C_{125}:C_5$	$5^{4}$	$5^{3}$	$2$	$2$	$4$	$2$	$145$	$20$	$16$	$15$	$C_{25}$	$C_5^2$	$C_5$	$C_{5} \times C_{25}$	$2^{2} \cdot 5^{4}$	$2^{2} \cdot 5^{2}$	$125$	$125$	$5$	$10$	$100$	$5$	$10$	$100$	Nilpotent - 2
625.9	$C_5.\He_5$	$5^{4}$	$5^{2}$	$3$	$2$	$4$	$2$	$49$	$70$	$18$	$10$	$C_5$	$\He_5$	$C_5^2$	$C_{5}^{2}$	$2^{3} \cdot 5^{4}$	$2^{3} \cdot 5$	$125$	$125$	$5$	$10$	$100$	$5$	$10$	$100$	Nilpotent - 3
625.10	$C_5.\He_5$	$5^{4}$	$5^{2}$	$3$	$2$	$4$	$2$	$49$	$70$	$18$	$10$	$C_5$	$\He_5$	$C_5^2$	$C_{5}^{2}$	$2^{3} \cdot 5^{4}$	$2^{3} \cdot 5$	$125$	$125$	$5$	$10$	$100$	$5$	$10$	$100$	Nilpotent - 3
729.8	$C_3^2.(C_3\times C_{27})$	$3^{6}$	$3^{3}$	$3$	$2$	$6$	$2$	$153$	$157$	$63$	$28$	$C_3\times C_9$	$\He_3$	$C_3^2$	$C_{3} \times C_{27}$	$2 \cdot 3^{7}$	$2 \cdot 3^{4}$	$243$	$36$	$-$	$-$	$-$	$4$	$8$	$24$	Nilpotent - 3
729.13	$C_3^3.(C_3\times C_9)$	$3^{6}$	$3^{3}$	$3$	$2$	$6$	$2$	$105$	$238$	$66$	$22$	$C_3\times C_9$	$\He_3$	$C_3^3$	$C_{3} \times C_{9}$	$2 \cdot 3^{7}$	$2 \cdot 3^{4}$	$243$	$36$	$-$	$-$	$-$	$6$	$12$	$24$	Nilpotent - 3
729.14	$C_3^3.(C_3\times C_9)$	$3^{6}$	$3^{3}$	$3$	$2$	$6$	$2$	$105$	$157$	$57$	$22$	$C_3\times C_9$	$\He_3$	$C_3^3$	$C_{3} \times C_{9}$	$2 \cdot 3^{7}$	$2 \cdot 3^{4}$	$243$	$36$	$-$	$-$	$-$	$6$	$12$	$24$	Nilpotent - 3

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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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