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 549 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
120.4	$C_{120}$	$2^{3} \cdot 3 \cdot 5$	$2^{3} \cdot 3 \cdot 5$	$1$	$1$	$5$	$1$	$120$	$16$	$16$	$16$	$C_{120}$	$C_1$	$C_1$	$C_{120}$	$2^{5}$	$2^{5}$	$120$	$16$	$1$	$2$	$32$	$1$	$2$	$10$	Cyclic
168.6	$C_{168}$	$2^{3} \cdot 3 \cdot 7$	$2^{3} \cdot 3 \cdot 7$	$1$	$1$	$5$	$1$	$168$	$16$	$16$	$16$	$C_{168}$	$C_1$	$C_1$	$C_{168}$	$2^{4} \cdot 3$	$2^{4} \cdot 3$	$168$	$18$	$1$	$2$	$48$	$1$	$2$	$12$	Cyclic
240.4	$C_{240}$	$2^{4} \cdot 3 \cdot 5$	$2^{4} \cdot 3 \cdot 5$	$1$	$1$	$6$	$1$	$240$	$20$	$20$	$20$	$C_{240}$	$C_1$	$C_1$	$C_{240}$	$2^{6}$	$2^{6}$	$240$	$24$	$1$	$2$	$64$	$1$	$2$	$14$	Cyclic
264.4	$C_{264}$	$2^{3} \cdot 3 \cdot 11$	$2^{3} \cdot 3 \cdot 11$	$1$	$1$	$5$	$1$	$264$	$16$	$16$	$16$	$C_{264}$	$C_1$	$C_1$	$C_{264}$	$2^{4} \cdot 5$	$2^{4} \cdot 5$	$264$	$22$	$1$	$2$	$80$	$1$	$2$	$16$	Cyclic
270.4	$C_{270}$	$2 \cdot 3^{3} \cdot 5$	$2 \cdot 3^{3} \cdot 5$	$1$	$1$	$5$	$1$	$270$	$16$	$16$	$16$	$C_{270}$	$C_1$	$C_1$	$C_{270}$	$2^{3} \cdot 3^{2}$	$2^{3} \cdot 3^{2}$	$270$	$34$	$1$	$2$	$72$	$1$	$2$	$22$	Cyclic
280.4	$C_{280}$	$2^{3} \cdot 5 \cdot 7$	$2^{3} \cdot 5 \cdot 7$	$1$	$1$	$5$	$1$	$280$	$16$	$16$	$16$	$C_{280}$	$C_1$	$C_1$	$C_{280}$	$2^{5} \cdot 3$	$2^{5} \cdot 3$	$280$	$20$	$1$	$2$	$96$	$1$	$2$	$14$	Cyclic
312.6	$C_{312}$	$2^{3} \cdot 3 \cdot 13$	$2^{3} \cdot 3 \cdot 13$	$1$	$1$	$5$	$1$	$312$	$16$	$16$	$16$	$C_{312}$	$C_1$	$C_1$	$C_{312}$	$2^{5} \cdot 3$	$2^{5} \cdot 3$	$312$	$24$	$1$	$2$	$96$	$1$	$2$	$18$	Cyclic
336.6	$C_{336}$	$2^{4} \cdot 3 \cdot 7$	$2^{4} \cdot 3 \cdot 7$	$1$	$1$	$6$	$1$	$336$	$20$	$20$	$20$	$C_{336}$	$C_1$	$C_1$	$C_{336}$	$2^{5} \cdot 3$	$2^{5} \cdot 3$	$336$	$26$	$1$	$2$	$96$	$1$	$2$	$16$	Cyclic
378.6	$C_{378}$	$2 \cdot 3^{3} \cdot 7$	$2 \cdot 3^{3} \cdot 7$	$1$	$1$	$5$	$1$	$378$	$16$	$16$	$16$	$C_{378}$	$C_1$	$C_1$	$C_{378}$	$2^{2} \cdot 3^{3}$	$2^{2} \cdot 3^{3}$	$378$	$36$	$1$	$2$	$108$	$1$	$2$	$24$	Cyclic
408.4	$C_{408}$	$2^{3} \cdot 3 \cdot 17$	$2^{3} \cdot 3 \cdot 17$	$1$	$1$	$5$	$1$	$408$	$16$	$16$	$16$	$C_{408}$	$C_1$	$C_1$	$C_{408}$	$2^{7}$	$2^{7}$	$408$	$28$	$1$	$2$	$128$	$1$	$2$	$22$	Cyclic
440.6	$C_{440}$	$2^{3} \cdot 5 \cdot 11$	$2^{3} \cdot 5 \cdot 11$	$1$	$1$	$5$	$1$	$440$	$16$	$16$	$16$	$C_{440}$	$C_1$	$C_1$	$C_{440}$	$2^{5} \cdot 5$	$2^{5} \cdot 5$	$440$	$24$	$1$	$2$	$160$	$1$	$2$	$18$	Cyclic
456.6	$C_{456}$	$2^{3} \cdot 3 \cdot 19$	$2^{3} \cdot 3 \cdot 19$	$1$	$1$	$5$	$1$	$456$	$16$	$16$	$16$	$C_{456}$	$C_1$	$C_1$	$C_{456}$	$2^{4} \cdot 3^{2}$	$2^{4} \cdot 3^{2}$	$456$	$30$	$1$	$2$	$144$	$1$	$2$	$24$	Cyclic
480.4	$C_{480}$	$2^{5} \cdot 3 \cdot 5$	$2^{5} \cdot 3 \cdot 5$	$1$	$1$	$7$	$1$	$480$	$24$	$24$	$24$	$C_{480}$	$C_1$	$C_1$	$C_{480}$	$2^{7}$	$2^{7}$	$480$	$40$	$1$	$2$	$128$	$1$	$2$	$22$	Cyclic
520.4	$C_{520}$	$2^{3} \cdot 5 \cdot 13$	$2^{3} \cdot 5 \cdot 13$	$1$	$1$	$5$	$1$	$520$	$16$	$16$	$16$	$C_{520}$	$C_1$	$C_1$	$C_{520}$	$2^{6} \cdot 3$	$2^{6} \cdot 3$	$520$	$26$	$1$	?	$192$	$1$	?	?	Cyclic
528.4	$C_{528}$	$2^{4} \cdot 3 \cdot 11$	$2^{4} \cdot 3 \cdot 11$	$1$	$1$	$6$	$1$	$528$	$20$	$20$	$20$	$C_{528}$	$C_1$	$C_1$	$C_{528}$	$2^{5} \cdot 5$	$2^{5} \cdot 5$	$528$	$30$	$1$	?	$160$	$1$	?	?	Cyclic
552.4	$C_{552}$	$2^{3} \cdot 3 \cdot 23$	$2^{3} \cdot 3 \cdot 23$	$1$	$1$	$5$	$1$	$552$	$16$	$16$	$16$	$C_{552}$	$C_1$	$C_1$	$C_{552}$	$2^{4} \cdot 11$	$2^{4} \cdot 11$	$552$	$34$	$1$	?	$176$	$1$	?	?	Cyclic
560.4	$C_{560}$	$2^{4} \cdot 5 \cdot 7$	$2^{4} \cdot 5 \cdot 7$	$1$	$1$	$6$	$1$	$560$	$20$	$20$	$20$	$C_{560}$	$C_1$	$C_1$	$C_{560}$	$2^{6} \cdot 3$	$2^{6} \cdot 3$	$560$	$28$	$1$	?	$192$	$1$	?	?	Cyclic
594.4	$C_{594}$	$2 \cdot 3^{3} \cdot 11$	$2 \cdot 3^{3} \cdot 11$	$1$	$1$	$5$	$1$	$594$	$16$	$16$	$16$	$C_{594}$	$C_1$	$C_1$	$C_{594}$	$2^{2} \cdot 3^{2} \cdot 5$	$2^{2} \cdot 3^{2} \cdot 5$	$594$	$40$	$1$	?	$180$	$1$	?	?	Cyclic
616.4	$C_{616}$	$2^{3} \cdot 7 \cdot 11$	$2^{3} \cdot 7 \cdot 11$	$1$	$1$	$5$	$1$	$616$	$16$	$16$	$16$	$C_{616}$	$C_1$	$C_1$	$C_{616}$	$2^{4} \cdot 3 \cdot 5$	$2^{4} \cdot 3 \cdot 5$	$616$	$26$	$1$	?	$240$	$1$	?	?	Cyclic
624.6	$C_{624}$	$2^{4} \cdot 3 \cdot 13$	$2^{4} \cdot 3 \cdot 13$	$1$	$1$	$6$	$1$	$624$	$20$	$20$	$20$	$C_{624}$	$C_1$	$C_1$	$C_{624}$	$2^{6} \cdot 3$	$2^{6} \cdot 3$	$624$	$32$	$1$	?	$192$	$1$	?	?	Cyclic
672.6	$C_{672}$	$2^{5} \cdot 3 \cdot 7$	$2^{5} \cdot 3 \cdot 7$	$1$	$1$	$7$	$1$	$672$	$24$	$24$	$24$	$C_{672}$	$C_1$	$C_1$	$C_{672}$	$2^{6} \cdot 3$	$2^{6} \cdot 3$	$672$	$42$	$1$	?	$192$	$1$	?	?	Cyclic
680.4	$C_{680}$	$2^{3} \cdot 5 \cdot 17$	$2^{3} \cdot 5 \cdot 17$	$1$	$1$	$5$	$1$	$680$	$16$	$16$	$16$	$C_{680}$	$C_1$	$C_1$	$C_{680}$	$2^{8}$	$2^{8}$	$680$	$30$	$1$	?	$256$	$1$	?	?	Cyclic
696.4	$C_{696}$	$2^{3} \cdot 3 \cdot 29$	$2^{3} \cdot 3 \cdot 29$	$1$	$1$	$5$	$1$	$696$	$16$	$16$	$16$	$C_{696}$	$C_1$	$C_1$	$C_{696}$	$2^{5} \cdot 7$	$2^{5} \cdot 7$	$696$	$40$	$1$	?	$224$	$1$	?	?	Cyclic
702.6	$C_{702}$	$2 \cdot 3^{3} \cdot 13$	$2 \cdot 3^{3} \cdot 13$	$1$	$1$	$5$	$1$	$702$	$16$	$16$	$16$	$C_{702}$	$C_1$	$C_1$	$C_{702}$	$2^{3} \cdot 3^{3}$	$2^{3} \cdot 3^{3}$	$702$	$42$	$1$	?	$216$	$1$	?	?	Cyclic
728.4	$C_{728}$	$2^{3} \cdot 7 \cdot 13$	$2^{3} \cdot 7 \cdot 13$	$1$	$1$	$5$	$1$	$728$	$16$	$16$	$16$	$C_{728}$	$C_1$	$C_1$	$C_{728}$	$2^{5} \cdot 3^{2}$	$2^{5} \cdot 3^{2}$	$728$	$28$	$1$	?	$288$	$1$	?	?	Cyclic
744.6	$C_{744}$	$2^{3} \cdot 3 \cdot 31$	$2^{3} \cdot 3 \cdot 31$	$1$	$1$	$5$	$1$	$744$	$16$	$16$	$16$	$C_{744}$	$C_1$	$C_1$	$C_{744}$	$2^{4} \cdot 3 \cdot 5$	$2^{4} \cdot 3 \cdot 5$	$744$	$42$	$1$	?	$240$	$1$	?	?	Cyclic
750.4	$C_{750}$	$2 \cdot 3 \cdot 5^{3}$	$2 \cdot 3 \cdot 5^{3}$	$1$	$1$	$5$	$1$	$750$	$16$	$16$	$16$	$C_{750}$	$C_1$	$C_1$	$C_{750}$	$2^{3} \cdot 5^{2}$	$2^{3} \cdot 5^{2}$	$750$	$130$	$1$	?	$200$	$1$	?	?	Cyclic
760.4	$C_{760}$	$2^{3} \cdot 5 \cdot 19$	$2^{3} \cdot 5 \cdot 19$	$1$	$1$	$5$	$1$	$760$	$16$	$16$	$16$	$C_{760}$	$C_1$	$C_1$	$C_{760}$	$2^{5} \cdot 3^{2}$	$2^{5} \cdot 3^{2}$	$760$	$32$	$1$	?	$288$	$1$	?	?	Cyclic
810.4	$C_{810}$	$2 \cdot 3^{4} \cdot 5$	$2 \cdot 3^{4} \cdot 5$	$1$	$1$	$6$	$1$	$810$	$20$	$20$	$20$	$C_{810}$	$C_1$	$C_1$	$C_{810}$	$2^{3} \cdot 3^{3}$	$2^{3} \cdot 3^{3}$	$810$	$88$	$1$	?	$216$	$1$	?	?	Cyclic
816.4	$C_{816}$	$2^{4} \cdot 3 \cdot 17$	$2^{4} \cdot 3 \cdot 17$	$1$	$1$	$6$	$1$	$816$	$20$	$20$	$20$	$C_{816}$	$C_1$	$C_1$	$C_{816}$	$2^{8}$	$2^{8}$	$816$	$36$	$1$	?	$256$	$1$	?	?	Cyclic
840.12	$C_{840}$	$2^{3} \cdot 3 \cdot 5 \cdot 7$	$2^{3} \cdot 3 \cdot 5 \cdot 7$	$1$	$1$	$6$	$1$	$840$	$32$	$32$	$32$	$C_{840}$	$C_1$	$C_1$	$C_{840}$	$2^{6} \cdot 3$	$2^{6} \cdot 3$	$840$	$23$	$1$	?	$192$	$1$	?	?	Cyclic
880.6	$C_{880}$	$2^{4} \cdot 5 \cdot 11$	$2^{4} \cdot 5 \cdot 11$	$1$	$1$	$6$	$1$	$880$	$20$	$20$	$20$	$C_{880}$	$C_1$	$C_1$	$C_{880}$	$2^{6} \cdot 5$	$2^{6} \cdot 5$	$880$	$32$	$1$	?	$320$	$1$	?	?	Cyclic
888.6	$C_{888}$	$2^{3} \cdot 3 \cdot 37$	$2^{3} \cdot 3 \cdot 37$	$1$	$1$	$5$	$1$	$888$	$16$	$16$	$16$	$C_{888}$	$C_1$	$C_1$	$C_{888}$	$2^{5} \cdot 3^{2}$	$2^{5} \cdot 3^{2}$	$888$	$48$	$1$	?	$288$	$1$	?	?	Cyclic
912.6	$C_{912}$	$2^{4} \cdot 3 \cdot 19$	$2^{4} \cdot 3 \cdot 19$	$1$	$1$	$6$	$1$	$912$	$20$	$20$	$20$	$C_{912}$	$C_1$	$C_1$	$C_{912}$	$2^{5} \cdot 3^{2}$	$2^{5} \cdot 3^{2}$	$912$	$38$	$1$	?	$288$	$1$	?	?	Cyclic
918.4	$C_{918}$	$2 \cdot 3^{3} \cdot 17$	$2 \cdot 3^{3} \cdot 17$	$1$	$1$	$5$	$1$	$918$	$16$	$16$	$16$	$C_{918}$	$C_1$	$C_1$	$C_{918}$	$2^{5} \cdot 3^{2}$	$2^{5} \cdot 3^{2}$	$918$	$46$	$1$	?	$288$	$1$	?	?	Cyclic
920.4	$C_{920}$	$2^{3} \cdot 5 \cdot 23$	$2^{3} \cdot 5 \cdot 23$	$1$	$1$	$5$	$1$	$920$	$16$	$16$	$16$	$C_{920}$	$C_1$	$C_1$	$C_{920}$	$2^{5} \cdot 11$	$2^{5} \cdot 11$	$920$	$36$	$1$	?	$352$	$1$	?	?	Cyclic
945.2	$C_{945}$	$3^{3} \cdot 5 \cdot 7$	$3^{3} \cdot 5 \cdot 7$	$1$	$1$	$5$	$1$	$945$	$16$	$16$	$16$	$C_{945}$	$C_1$	$C_1$	$C_{945}$	$2^{4} \cdot 3^{3}$	$2^{4} \cdot 3^{3}$	$945$	$39$	$1$	?	?	$1$	?	?	Cyclic
952.4	$C_{952}$	$2^{3} \cdot 7 \cdot 17$	$2^{3} \cdot 7 \cdot 17$	$1$	$1$	$5$	$1$	$952$	$16$	$16$	$16$	$C_{952}$	$C_1$	$C_1$	$C_{952}$	$2^{7} \cdot 3$	$2^{7} \cdot 3$	$952$	$32$	$1$	?	?	$1$	?	?	Cyclic
960.4	$C_{960}$	$2^{6} \cdot 3 \cdot 5$	$2^{6} \cdot 3 \cdot 5$	$1$	$1$	$8$	$1$	$960$	$28$	$28$	$28$	$C_{960}$	$C_1$	$C_1$	$C_{960}$	$2^{8}$	$2^{8}$	$960$	$72$	$1$	?	$256$	$1$	?	?	Cyclic
984.4	$C_{984}$	$2^{3} \cdot 3 \cdot 41$	$2^{3} \cdot 3 \cdot 41$	$1$	$1$	$5$	$1$	$984$	$16$	$16$	$16$	$C_{984}$	$C_1$	$C_1$	$C_{984}$	$2^{6} \cdot 5$	$2^{6} \cdot 5$	$984$	$52$	$1$	?	$320$	$1$	?	?	Cyclic
1026.6	$C_{1026}$	$2 \cdot 3^{3} \cdot 19$	$2 \cdot 3^{3} \cdot 19$	$1$	$1$	$5$	$1$	$1026$	$16$	$16$	$16$	$C_{1026}$	$C_1$	$C_1$	$C_{1026}$	$2^{2} \cdot 3^{4}$	$2^{2} \cdot 3^{4}$	$1026$	$48$	$1$	?	$324$	$1$	?	?	Cyclic
1032.6	$C_{1032}$	$2^{3} \cdot 3 \cdot 43$	$2^{3} \cdot 3 \cdot 43$	$1$	$1$	$5$	$1$	$1032$	$16$	$16$	$16$	$C_{1032}$	$C_1$	$C_1$	$C_{1032}$	$2^{4} \cdot 3 \cdot 7$	$2^{4} \cdot 3 \cdot 7$	$1032$	$54$	$1$	?	?	$1$	?	?	Cyclic
1040.4	$C_{1040}$	$2^{4} \cdot 5 \cdot 13$	$2^{4} \cdot 5 \cdot 13$	$1$	$1$	$6$	$1$	$1040$	$20$	$20$	$20$	$C_{1040}$	$C_1$	$C_1$	$C_{1040}$	$2^{7} \cdot 3$	$2^{7} \cdot 3$	$1040$	$34$	$1$	?	?	$1$	?	?	Cyclic
1056.4	$C_{1056}$	$2^{5} \cdot 3 \cdot 11$	$2^{5} \cdot 3 \cdot 11$	$1$	$1$	$7$	$1$	$1056$	$24$	$24$	$24$	$C_{1056}$	$C_1$	$C_1$	$C_{1056}$	$2^{6} \cdot 5$	$2^{6} \cdot 5$	$1056$	$46$	$1$	?	$320$	$1$	?	?	Cyclic
1064.4	$C_{1064}$	$2^{3} \cdot 7 \cdot 19$	$2^{3} \cdot 7 \cdot 19$	$1$	$1$	$5$	$1$	$1064$	$16$	$16$	$16$	$C_{1064}$	$C_1$	$C_1$	$C_{1064}$	$2^{4} \cdot 3^{3}$	$2^{4} \cdot 3^{3}$	$1064$	$34$	$1$	?	?	$1$	?	?	Cyclic
1104.4	$C_{1104}$	$2^{4} \cdot 3 \cdot 23$	$2^{4} \cdot 3 \cdot 23$	$1$	$1$	$6$	$1$	$1104$	$20$	$20$	$20$	$C_{1104}$	$C_1$	$C_1$	$C_{1104}$	$2^{5} \cdot 11$	$2^{5} \cdot 11$	$1104$	$42$	$1$	?	?	$1$	?	?	Cyclic
1120.4	$C_{1120}$	$2^{5} \cdot 5 \cdot 7$	$2^{5} \cdot 5 \cdot 7$	$1$	$1$	$7$	$1$	$1120$	$24$	$24$	$24$	$C_{1120}$	$C_1$	$C_1$	$C_{1120}$	$2^{7} \cdot 3$	$2^{7} \cdot 3$	$1120$	$44$	$1$	?	?	$1$	?	?	Cyclic
1128.4	$C_{1128}$	$2^{3} \cdot 3 \cdot 47$	$2^{3} \cdot 3 \cdot 47$	$1$	$1$	$5$	$1$	$1128$	$16$	$16$	$16$	$C_{1128}$	$C_1$	$C_1$	$C_{1128}$	$2^{4} \cdot 23$	$2^{4} \cdot 23$	$1128$	$58$	$1$	?	?	$1$	?	?	Cyclic
1134.6	$C_{1134}$	$2 \cdot 3^{4} \cdot 7$	$2 \cdot 3^{4} \cdot 7$	$1$	$1$	$6$	$1$	$1134$	$20$	$20$	$20$	$C_{1134}$	$C_1$	$C_1$	$C_{1134}$	$2^{2} \cdot 3^{4}$	$2^{2} \cdot 3^{4}$	$1134$	$90$	$1$	?	$324$	$1$	?	?	Cyclic
1144.4	$C_{1144}$	$2^{3} \cdot 11 \cdot 13$	$2^{3} \cdot 11 \cdot 13$	$1$	$1$	$5$	$1$	$1144$	$16$	$16$	$16$	$C_{1144}$	$C_1$	$C_1$	$C_{1144}$	$2^{5} \cdot 3 \cdot 5$	$2^{5} \cdot 3 \cdot 5$	$1144$	$32$	$1$	?	?	$1$	?	?	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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