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 84469 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
1501.1	$C_{1501}$	$19 \cdot 79$	$19 \cdot 79$	$1$	$1$	$2$	$1$	$1501$	$4$	$4$	$4$	$C_{1501}$	$C_1$	$C_1$	$C_{1501}$	$2^{2} \cdot 3^{3} \cdot 13$	$2^{2} \cdot 3^{3} \cdot 13$	$1501$	$98$	$1$	?	?	$1$	?	?	Cyclic
1502.1	$D_{751}$	$2 \cdot 751$	$2 \cdot 751$	$-1$	$2$	$2$	$2$	$377$	$754$	$4$	$3$	$C_1$	$D_{751}$	$C_{751}$	$C_{2}$	$2 \cdot 3 \cdot 5^{3} \cdot 751$	$3 \cdot 5^{3}$	$751$	$751$	$2$	?	?	?	?	?	Solvable - 2
1502.2	$C_{1502}$	$2 \cdot 751$	$2 \cdot 751$	$1$	$1$	$2$	$1$	$1502$	$4$	$4$	$4$	$C_{1502}$	$C_1$	$C_1$	$C_{1502}$	$2 \cdot 3 \cdot 5^{3}$	$2 \cdot 3 \cdot 5^{3}$	$1502$	$753$	$1$	?	?	$1$	?	?	Cyclic
1503.1	$C_{1503}$	$3^{2} \cdot 167$	$3^{2} \cdot 167$	$1$	$1$	$3$	$1$	$1503$	$6$	$6$	$6$	$C_{1503}$	$C_1$	$C_1$	$C_{1503}$	$2^{2} \cdot 3 \cdot 83$	$2^{2} \cdot 3 \cdot 83$	$1503$	$176$	$1$	?	?	$1$	?	?	Cyclic
1503.2	$C_3\times C_{501}$	$3^{2} \cdot 167$	$3 \cdot 167$	$1$	$1$	$3$	$2$	$1503$	$12$	$12$	$12$	$C_3\times C_{501}$	$C_1$	$C_1$	$C_{3} \times C_{501}$	$2^{5} \cdot 3 \cdot 83$	$2^{5} \cdot 3 \cdot 83$	$1503$	$173$	$-$	?	$-$	$2$	?	?	Abelian - 2
1504.1	$C_{1504}$	$2^{5} \cdot 47$	$2^{5} \cdot 47$	$1$	$1$	$6$	$1$	$1504$	$12$	$12$	$12$	$C_{1504}$	$C_1$	$C_1$	$C_{1504}$	$2^{5} \cdot 23$	$2^{5} \cdot 23$	$1504$	$79$	$1$	?	?	$1$	?	?	Cyclic
1504.2	$C_{94}.C_4^2$	$2^{5} \cdot 47$	$2^{2} \cdot 47$	$2$	$2$	$6$	$2$	$940$	$100$	$76$	$52$	$C_2^2\times C_{94}$	$C_2^2$	$C_2$	$C_{4} \times C_{188}$	$2^{8} \cdot 3 \cdot 23$	$2^{6} \cdot 3 \cdot 23$	$1504$	$59$	$-$	?	$-$	?	?	?	Nilpotent - 2
1504.3	$C_4\times C_{376}$	$2^{5} \cdot 47$	$2^{3} \cdot 47$	$1$	$1$	$6$	$2$	$1504$	$44$	$44$	$44$	$C_4\times C_{376}$	$C_1$	$C_1$	$C_{4} \times C_{376}$	$2^{8} \cdot 23$	$2^{8} \cdot 23$	$1504$	$59$	$-$	?	$-$	$2$	?	?	Abelian - 2
1504.4	$C_8:C_{188}$	$2^{5} \cdot 47$	$2^{3} \cdot 47$	$2$	$2$	$6$	$2$	$940$	$44$	$40$	$36$	$C_2\times C_{188}$	$C_2^2$	$C_2$	$C_{4} \times C_{188}$	$2^{8} \cdot 23$	$2^{6} \cdot 23$	$1504$	$59$	$-$	?	$-$	?	?	?	Nilpotent - 2
1504.5	$C_2^2:C_{376}$	$2^{5} \cdot 47$	$2^{3} \cdot 47$	$2$	$2$	$6$	$2$	$940$	$68$	$50$	$32$	$C_2\times C_{188}$	$C_2^2$	$C_2$	$C_{2} \times C_{376}$	$2^{7} \cdot 23$	$2^{5} \cdot 23$	$752$	$59$	$-$	?	$-$	?	?	?	Nilpotent - 2
1504.6	$C_2^3:C_{188}$	$2^{5} \cdot 47$	$2^{2} \cdot 47$	$3$	$2$	$6$	$2$	$517$	$100$	$52$	$24$	$C_{94}$	$C_2^2:C_4$	$C_2^2$	$C_{2} \times C_{188}$	$2^{7} \cdot 23$	$2^{3} \cdot 23$	$376$	$55$	$4$	?	$184$	?	?	?	Nilpotent - 3
1504.7	$\OD_{16}:C_{94}$	$2^{5} \cdot 47$	$2^{3} \cdot 47$	$3$	$2$	$6$	$2$	$517$	$84$	$46$	$24$	$C_{94}$	$C_2^2:C_4$	$C_2^2$	$C_{2} \times C_{188}$	$2^{8} \cdot 23$	$2^{4} \cdot 23$	$376$	$55$	$4$	?	$184$	?	?	?	Nilpotent - 3
1504.8	$C_{188}.D_4$	$2^{5} \cdot 47$	$2^{3} \cdot 47$	$3$	$2$	$6$	$2$	$517$	$52$	$38$	$24$	$C_{94}$	$C_2^2:C_4$	$C_2^2$	$C_{2} \times C_{188}$	$2^{8} \cdot 23$	$2^{4} \cdot 23$	$752$	$63$	$4$	?	$368$	?	?	?	Nilpotent - 3
1504.9	$D_4:C_{188}$	$2^{5} \cdot 47$	$2^{3} \cdot 47$	$3$	$2$	$6$	$2$	$658$	$92$	$50$	$28$	$C_2\times C_{94}$	$D_4$	$C_4$	$C_{2} \times C_{188}$	$2^{7} \cdot 23$	$2^{4} \cdot 23$	$752$	$59$	$-$	?	$-$	?	?	?	Nilpotent - 3
1504.10	$Q_8:C_{188}$	$2^{5} \cdot 47$	$2^{3} \cdot 47$	$3$	$2$	$6$	$2$	$658$	$60$	$42$	$28$	$C_2\times C_{94}$	$D_4$	$C_4$	$C_{2} \times C_{188}$	$2^{7} \cdot 23$	$2^{4} \cdot 23$	$1504$	$59$	$-$	?	$-$	?	?	?	Nilpotent - 3
1504.11	$D_4:C_{188}$	$2^{5} \cdot 47$	$2^{3} \cdot 47$	$3$	$2$	$6$	$2$	$658$	$68$	$44$	$24$	$C_{188}$	$D_4$	$C_4$	$C_{2} \times C_{188}$	$2^{6} \cdot 23$	$2^{3} \cdot 23$	$376$	$55$	$2$	?	$184$	?	?	?	Nilpotent - 3
1504.12	$C_4:C_{376}$	$2^{5} \cdot 47$	$2^{3} \cdot 47$	$2$	$2$	$6$	$2$	$940$	$44$	$38$	$32$	$C_2\times C_{188}$	$C_2^2$	$C_2$	$C_{2} \times C_{376}$	$2^{7} \cdot 23$	$2^{5} \cdot 23$	$1504$	$59$	$-$	?	$-$	?	?	?	Nilpotent - 2
1504.13	$C_8:C_{188}$	$2^{5} \cdot 47$	$2^{3} \cdot 47$	$3$	$2$	$6$	$2$	$658$	$52$	$36$	$28$	$C_2\times C_{94}$	$D_4$	$C_4$	$C_{2} \times C_{188}$	$2^{8} \cdot 23$	$2^{5} \cdot 23$	$1504$	$59$	$-$	?	$-$	?	?	?	Nilpotent - 3
1504.14	$C_8:C_{188}$	$2^{5} \cdot 47$	$2^{3} \cdot 47$	$3$	$2$	$6$	$2$	$658$	$52$	$36$	$28$	$C_2\times C_{94}$	$D_4$	$C_4$	$C_{2} \times C_{188}$	$2^{8} \cdot 23$	$2^{5} \cdot 23$	$1504$	$59$	$-$	?	$-$	?	?	?	Nilpotent - 3
1504.15	$C_8.C_{188}$	$2^{5} \cdot 47$	$2^{3} \cdot 47$	$3$	$2$	$6$	$2$	$658$	$36$	$30$	$24$	$C_{188}$	$D_4$	$C_4$	$C_{2} \times C_{188}$	$2^{7} \cdot 23$	$2^{4} \cdot 23$	$752$	$63$	$2$	?	$368$	?	?	?	Nilpotent - 3
1504.16	$C_2\times C_{752}$	$2^{5} \cdot 47$	$2^{4} \cdot 47$	$1$	$1$	$6$	$2$	$1504$	$28$	$28$	$28$	$C_2\times C_{752}$	$C_1$	$C_1$	$C_{2} \times C_{752}$	$2^{6} \cdot 23$	$2^{6} \cdot 23$	$1504$	$65$	$-$	?	$-$	$2$	?	?	Abelian - 2
1504.17	$\OD_{32}\times C_{47}$	$2^{5} \cdot 47$	$2^{4} \cdot 47$	$2$	$2$	$6$	$2$	$940$	$28$	$26$	$24$	$C_{376}$	$C_2^2$	$C_2$	$C_{2} \times C_{376}$	$2^{6} \cdot 23$	$2^{4} \cdot 23$	$752$	$63$	$2$	?	$368$	?	?	?	Nilpotent - 2
1504.18	$D_{16}\times C_{47}$	$2^{5} \cdot 47$	$2^{4} \cdot 47$	$4$	$2$	$6$	$2$	$517$	$72$	$28$	$16$	$C_{94}$	$D_8$	$C_8$	$C_{2} \times C_{94}$	$2^{8} \cdot 23$	$2^{4} \cdot 23$	$752$	$63$	$2$	?	$368$	?	?	?	Nilpotent - 4
1504.19	$\SD_{32}\times C_{47}$	$2^{5} \cdot 47$	$2^{4} \cdot 47$	$4$	$2$	$6$	$2$	$517$	$56$	$26$	$16$	$C_{94}$	$D_8$	$C_8$	$C_{2} \times C_{94}$	$2^{7} \cdot 23$	$2^{3} \cdot 23$	$752$	$63$	$2$	?	$368$	?	?	?	Nilpotent - 4
1504.20	$Q_{32}\times C_{47}$	$2^{5} \cdot 47$	$2^{4} \cdot 47$	$4$	$2$	$6$	$2$	$517$	$40$	$24$	$16$	$C_{94}$	$D_8$	$C_8$	$C_{2} \times C_{94}$	$2^{8} \cdot 23$	$2^{4} \cdot 23$	$1504$	$79$	$2$	?	$368$	?	?	?	Nilpotent - 4
1504.21	$C_2\times C_4\times C_{188}$	$2^{5} \cdot 47$	$2^{2} \cdot 47$	$1$	$1$	$6$	$3$	$1504$	$108$	$108$	$108$	$C_2\times C_4\times C_{188}$	$C_1$	$C_1$	$C_{2} \times C_{4} \times C_{188}$	$2^{10} \cdot 3 \cdot 23$	$2^{10} \cdot 3 \cdot 23$	$1504$	$57$	$-$	?	$-$	$3$	?	?	Abelian - 3
1504.22	$C_2^3:C_{188}$	$2^{5} \cdot 47$	$2^{2} \cdot 47$	$2$	$2$	$6$	$3$	$940$	$188$	$132$	$76$	$C_2^2\times C_{94}$	$C_2^2$	$C_2$	$C_{2}^{2} \times C_{188}$	$2^{10} \cdot 23$	$2^{8} \cdot 23$	$752$	$57$	$-$	?	$-$	?	?	?	Nilpotent - 2
1504.23	$C_2\times C_4:C_{188}$	$2^{5} \cdot 47$	$2^{2} \cdot 47$	$2$	$2$	$6$	$3$	$940$	$108$	$92$	$76$	$C_2^2\times C_{94}$	$C_2^2$	$C_2$	$C_{2}^{2} \times C_{188}$	$2^{10} \cdot 23$	$2^{8} \cdot 23$	$1504$	$57$	$-$	?	$-$	?	?	?	Nilpotent - 2
1504.24	$C_4^2:C_{94}$	$2^{5} \cdot 47$	$2^{2} \cdot 47$	$2$	$2$	$6$	$3$	$940$	$92$	$76$	$60$	$C_2\times C_{188}$	$C_2^2$	$C_2$	$C_{2}^{2} \times C_{188}$	$2^{9} \cdot 23$	$2^{7} \cdot 23$	$752$	$59$	$-$	?	$-$	?	?	?	Nilpotent - 2
1504.25	$D_4\times C_{188}$	$2^{5} \cdot 47$	$2^{2} \cdot 47$	$2$	$2$	$6$	$3$	$940$	$124$	$94$	$64$	$C_2\times C_{188}$	$C_2^2$	$C_2$	$C_{2}^{2} \times C_{188}$	$2^{8} \cdot 23$	$2^{6} \cdot 23$	$752$	$55$	$-$	?	$-$	?	?	?	Nilpotent - 2
1504.26	$Q_8\times C_{188}$	$2^{5} \cdot 47$	$2^{2} \cdot 47$	$2$	$2$	$6$	$3$	$940$	$76$	$70$	$64$	$C_2\times C_{188}$	$C_2^2$	$C_2$	$C_{2}^{2} \times C_{188}$	$2^{8} \cdot 3 \cdot 23$	$2^{6} \cdot 3 \cdot 23$	$1504$	$59$	$-$	?	$-$	?	?	?	Nilpotent - 2
1504.27	$C_2^4:C_{94}$	$2^{5} \cdot 47$	$2^{2} \cdot 47$	$2$	$2$	$6$	$3$	$658$	$212$	$130$	$52$	$C_2\times C_{94}$	$C_2^3$	$C_2^2$	$C_{2}^{2} \times C_{94}$	$2^{8} \cdot 3 \cdot 23$	$2^{5} \cdot 3 \cdot 23$	$376$	$55$	$-$	?	$-$	?	?	?	Nilpotent - 2
1504.28	$C_{188}:D_4$	$2^{5} \cdot 47$	$2^{2} \cdot 47$	$2$	$2$	$6$	$3$	$658$	$148$	$94$	$48$	$C_2\times C_{94}$	$C_2^3$	$C_2^2$	$C_{2}^{2} \times C_{94}$	$2^{8} \cdot 23$	$2^{5} \cdot 23$	$752$	$55$	$-$	?	$-$	?	?	?	Nilpotent - 2
1504.29	$C_{188}.D_4$	$2^{5} \cdot 47$	$2^{2} \cdot 47$	$2$	$2$	$6$	$3$	$658$	$100$	$74$	$48$	$C_2\times C_{94}$	$C_2^3$	$C_2^2$	$C_{2}^{2} \times C_{94}$	$2^{8} \cdot 23$	$2^{5} \cdot 23$	$752$	$59$	$-$	?	$-$	?	?	?	Nilpotent - 2
1504.30	$(C_2^2\times C_4):C_{94}$	$2^{5} \cdot 47$	$2^{2} \cdot 47$	$2$	$2$	$6$	$3$	$658$	$116$	$78$	$44$	$C_2\times C_{94}$	$C_2^3$	$C_2^2$	$C_{2}^{2} \times C_{94}$	$2^{8} \cdot 23$	$2^{5} \cdot 23$	$752$	$59$	$-$	?	$-$	?	?	?	Nilpotent - 2
1504.31	$C_4^2:C_{94}$	$2^{5} \cdot 47$	$2^{2} \cdot 47$	$2$	$2$	$6$	$3$	$658$	$116$	$76$	$44$	$C_2\times C_{94}$	$C_2^3$	$C_2^2$	$C_{2}^{2} \times C_{94}$	$2^{9} \cdot 23$	$2^{6} \cdot 23$	$752$	$59$	$-$	?	$-$	?	?	?	Nilpotent - 2
1504.32	$C_4^2.C_{94}$	$2^{5} \cdot 47$	$2^{2} \cdot 47$	$2$	$2$	$6$	$3$	$658$	$68$	$56$	$44$	$C_2\times C_{94}$	$C_2^3$	$C_2^2$	$C_{2}^{2} \times C_{94}$	$2^{9} \cdot 23$	$2^{6} \cdot 23$	$1504$	$63$	$-$	?	$-$	?	?	?	Nilpotent - 2
1504.33	$C_4^2:C_{94}$	$2^{5} \cdot 47$	$2^{2} \cdot 47$	$2$	$2$	$6$	$3$	$658$	$84$	$60$	$40$	$C_2\times C_{94}$	$C_2^3$	$C_2^2$	$C_{2}^{2} \times C_{94}$	$2^{7} \cdot 3 \cdot 23$	$2^{4} \cdot 3 \cdot 23$	$752$	$63$	$-$	?	$-$	?	?	?	Nilpotent - 2
1504.34	$C_4^2:C_{94}$	$2^{5} \cdot 47$	$2^{2} \cdot 47$	$2$	$2$	$6$	$3$	$658$	$180$	$108$	$52$	$C_2\times C_{94}$	$C_2^3$	$C_2^2$	$C_{2}^{2} \times C_{94}$	$2^{10} \cdot 3 \cdot 23$	$2^{7} \cdot 3 \cdot 23$	$752$	$55$	$-$	?	$-$	?	?	?	Nilpotent - 2
1504.35	$C_{188}:Q_8$	$2^{5} \cdot 47$	$2^{2} \cdot 47$	$2$	$2$	$6$	$3$	$658$	$84$	$68$	$52$	$C_2\times C_{94}$	$C_2^3$	$C_2^2$	$C_{2}^{2} \times C_{94}$	$2^{10} \cdot 23$	$2^{7} \cdot 23$	$1504$	$59$	$-$	?	$-$	?	?	?	Nilpotent - 2
1504.36	$C_2^2\times C_{376}$	$2^{5} \cdot 47$	$2^{3} \cdot 47$	$1$	$1$	$6$	$3$	$1504$	$76$	$76$	$76$	$C_2^2\times C_{376}$	$C_1$	$C_1$	$C_{2}^{2} \times C_{376}$	$2^{8} \cdot 3 \cdot 23$	$2^{8} \cdot 3 \cdot 23$	$1504$	$59$	$-$	?	$-$	$3$	?	?	Abelian - 3
1504.37	$\OD_{16}\times C_{94}$	$2^{5} \cdot 47$	$2^{3} \cdot 47$	$2$	$2$	$6$	$3$	$940$	$76$	$68$	$60$	$C_2\times C_{188}$	$C_2^2$	$C_2$	$C_{2}^{2} \times C_{188}$	$2^{8} \cdot 23$	$2^{6} \cdot 23$	$752$	$57$	$-$	?	$-$	?	?	?	Nilpotent - 2
1504.38	$\OD_{16}:C_{94}$	$2^{5} \cdot 47$	$2^{3} \cdot 47$	$2$	$2$	$6$	$3$	$940$	$68$	$62$	$56$	$C_{376}$	$C_2^2$	$C_2$	$C_{2}^{2} \times C_{188}$	$2^{6} \cdot 3 \cdot 23$	$2^{4} \cdot 3 \cdot 23$	$752$	$63$	$2$	?	$368$	?	?	?	Nilpotent - 2
1504.39	$D_8\times C_{94}$	$2^{5} \cdot 47$	$2^{3} \cdot 47$	$3$	$2$	$6$	$3$	$658$	$140$	$76$	$44$	$C_2\times C_{94}$	$D_4$	$C_4$	$C_{2}^{2} \times C_{94}$	$2^{9} \cdot 23$	$2^{6} \cdot 23$	$752$	$57$	$-$	?	$-$	?	?	?	Nilpotent - 3
1504.40	$\SD_{16}\times C_{94}$	$2^{5} \cdot 47$	$2^{3} \cdot 47$	$3$	$2$	$6$	$3$	$658$	$108$	$68$	$44$	$C_2\times C_{94}$	$D_4$	$C_4$	$C_{2}^{2} \times C_{94}$	$2^{8} \cdot 23$	$2^{5} \cdot 23$	$752$	$57$	$-$	?	$-$	?	?	?	Nilpotent - 3
1504.41	$Q_{16}\times C_{94}$	$2^{5} \cdot 47$	$2^{3} \cdot 47$	$3$	$2$	$6$	$3$	$658$	$76$	$60$	$44$	$C_2\times C_{94}$	$D_4$	$C_4$	$C_{2}^{2} \times C_{94}$	$2^{9} \cdot 23$	$2^{6} \cdot 23$	$1504$	$65$	$-$	?	$-$	?	?	?	Nilpotent - 3
1504.42	$D_8:C_{94}$	$2^{5} \cdot 47$	$2^{3} \cdot 47$	$3$	$2$	$6$	$3$	$658$	$92$	$62$	$40$	$C_{188}$	$D_4$	$C_4$	$C_{2}^{2} \times C_{94}$	$2^{7} \cdot 23$	$2^{4} \cdot 23$	$752$	$63$	$2$	?	$368$	?	?	?	Nilpotent - 3
1504.43	$D_8:C_{94}$	$2^{5} \cdot 47$	$2^{3} \cdot 47$	$3$	$2$	$6$	$3$	$517$	$116$	$68$	$40$	$C_{94}$	$C_2\times D_4$	$C_4$	$C_{2}^{2} \times C_{94}$	$2^{7} \cdot 23$	$2^{3} \cdot 23$	$376$	$55$	$4$	?	$184$	?	?	?	Nilpotent - 3
1504.44	$Q_{16}:C_{94}$	$2^{5} \cdot 47$	$2^{3} \cdot 47$	$3$	$2$	$6$	$3$	$517$	$84$	$60$	$40$	$C_{94}$	$C_2\times D_4$	$C_4$	$C_{2}^{2} \times C_{94}$	$2^{7} \cdot 23$	$2^{3} \cdot 23$	$752$	$63$	$4$	?	$368$	?	?	?	Nilpotent - 3
1504.45	$C_2^3\times C_{188}$	$2^{5} \cdot 47$	$2^{2} \cdot 47$	$1$	$1$	$6$	$4$	$1504$	$236$	$236$	$236$	$C_2^3\times C_{188}$	$C_1$	$C_1$	$C_{2}^{3} \times C_{188}$	$2^{11} \cdot 3 \cdot 7 \cdot 23$	$2^{11} \cdot 3 \cdot 7 \cdot 23$	$1504$	$57$	$-$	?	$-$	$4$	?	?	Abelian - 4

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Sato-Tate groups
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Lattices

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Motives

Groups

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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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Results (1-50 of 84469 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.

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