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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 (21 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
120.34	$S_5$	$\POMinus(4,2)$	$2^{3} \cdot 3 \cdot 5$	$2^{2} \cdot 3 \cdot 5$	$-1$	$1$	$2$	$2$	7	$156$	$19$	$3$	$C_1$	$S_5$	$A_5$	$C_{2}$	$2^{3} \cdot 3 \cdot 5$	$1$	$5$	$5$	$4$	$4$	$4$	$4$	$4$	$4$	Non-Solvable - 2
720.763	$S_6$	$\POMinus(4,3)$	$2^{4} \cdot 3^{2} \cdot 5$	$2^{2} \cdot 3 \cdot 5$	$-1$	$1$	$2$	$2$	11	$1455$	$56$	$3$	$C_1$	$S_6$	$A_6$	$C_{2}$	$2^{5} \cdot 3^{2} \cdot 5$	$2$	$6$	$6$	$5$	$5$	$5$	$5$	$5$	$5$	Non-Solvable - 2
8160.a	$\SOMinus(4,4)$	$\POMinus(4,4)$	$2^{5} \cdot 3 \cdot 5 \cdot 17$	$2^{2} \cdot 3 \cdot 5 \cdot 17$	$-1$	$1$	$2$	$2$	16	$10002$	$47$	$3$	$C_1$	$\SOMinus(4,4)$	$\SL(2,16)$	$C_{2}$	$2^{6} \cdot 3 \cdot 5 \cdot 17$	$2$	$17$	$17$	$16$	$16$	$16$	$16$	$16$	$16$	Non-Solvable - 2
15600.d	$\POMinus(4,5)$	$\POMinus(4,5)$	$2^{4} \cdot 3 \cdot 5^{2} \cdot 13$	$2^{2} \cdot 3 \cdot 5 \cdot 13$	$-1$	$1$	$2$	$2$	18	$31726$	$103$	$3$	$C_1$	$\POMinus(4,5)$	$\PSL(2,25)$	$C_{2}$	$2^{5} \cdot 3 \cdot 5^{2} \cdot 13$	$2$	$26$	$26$	$13$	$13$	$13$	$13$	$13$	$13$	Non-Solvable - 2
51840.b	$\SO(5,3)$	$\POMinus(6,2)$	$2^{7} \cdot 3^{4} \cdot 5$	$2^{3} \cdot 3^{2} \cdot 5$	$-1$	$1$	$2$	$2$	25				$C_1$	$\SO(5,3)$	$\SU(4,2)$	$C_{2}$	$2^{7} \cdot 3^{4} \cdot 5$	$1$		$27$	$6$	?	?	?	?	?	Non-Solvable - 2
117600.d	$\POMinus(4,7)$	$\POMinus(4,7)$	$2^{5} \cdot 3 \cdot 5^{2} \cdot 7^{2}$	$2^{3} \cdot 3 \cdot 5^{2} \cdot 7$	$-1$	$1$	$2$	$2$	27				$C_1$	$\POMinus(4,7)$	$\PSL(2,49)$	$C_{2}$	$2^{6} \cdot 3 \cdot 5^{2} \cdot 7^{2}$	$2$		$50$	$25$	?	?	?	?	?	Non-Solvable - 2
524160.a	$\SOMinus(4,8)$	$\POMinus(4,8)$	$2^{7} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 13$	$2^{2} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 13$	$-1$	$1$	$2$	$2$	46	$885500$	$127$	$3$	$C_1$	$\SOMinus(4,8)$	$\SL(2,64)$	$C_{2}$	$2^{7} \cdot 3^{3} \cdot 5 \cdot 7 \cdot 13$	$3$	$65$	$65$	$64$	$64$	$64$	$64$	$64$	$64$	Non-Solvable - 2
531360.d	$\POMinus(4,9)$	$\POMinus(4,9)$	$2^{5} \cdot 3^{4} \cdot 5 \cdot 41$	$2^{3} \cdot 3 \cdot 5 \cdot 41$	$-1$	$1$	$2$	$2$	38	$1289128$	$187$	$3$	$C_1$	$\POMinus(4,9)$	$\PSL(2,81)$	$C_{2}$	$2^{7} \cdot 3^{4} \cdot 5 \cdot 41$	$2^{2}$	$82$	$82$	$41$	$41$	$41$	$41$	$41$	$41$	Non-Solvable - 2
1771440.d	$\POMinus(4,11)$	$\POMinus(4,11)$	$2^{4} \cdot 3 \cdot 5 \cdot 11^{2} \cdot 61$	$2^{2} \cdot 3 \cdot 5 \cdot 11 \cdot 61$	$-1$	$1$	$2$	$2$	51				$C_1$	$\POMinus(4,11)$	$\PSL(2,121)$	$C_{2}$	$2^{5} \cdot 3 \cdot 5 \cdot 11^{2} \cdot 61$	$2$		$122$	$61$	?	?	?	?	?	Non-Solvable - 2
4826640.d	$\POMinus(4,13)$	$\POMinus(4,13)$	$2^{4} \cdot 3 \cdot 5 \cdot 7 \cdot 13^{2} \cdot 17$	$2^{2} \cdot 3 \cdot 5 \cdot 7 \cdot 13 \cdot 17$	$-1$	$1$	$2$	$2$	66				$C_1$	$\POMinus(4,13)$	$\PSL(2,169)$	$C_{2}$	$2^{5} \cdot 3 \cdot 5 \cdot 7 \cdot 13^{2} \cdot 17$	$2$		$170$	$85$	?	?	?	?	?	Non-Solvable - 2
13063680.c	$\POMinus(6,3)$	$\POMinus(6,3)$	$2^{9} \cdot 3^{6} \cdot 5 \cdot 7$	$2^{3} \cdot 3^{2} \cdot 5 \cdot 7$	$-1$	$1$	$3$	$2$	59			$6$	$C_1$	$\POMinus(6,3)$	$\PSU(4,3)$	$C_{2}^{2}$	$2^{10} \cdot 3^{6} \cdot 5 \cdot 7$	$2$	$112$	$112$	$21$	$21$	$21$	?	?	?	Non-Solvable - 3
24137280.c	$\POMinus(4,17)$	$\POMinus(4,17)$	$2^{6} \cdot 3^{2} \cdot 5 \cdot 17^{2} \cdot 29$	$2^{4} \cdot 3^{2} \cdot 5 \cdot 17 \cdot 29$	$-1$	$1$	$2$	$2$	102				$C_1$	$\POMinus(4,17)$	$\PSL(2,289)$	$C_{2}$	$2^{7} \cdot 3^{2} \cdot 5 \cdot 17^{2} \cdot 29$	$2$		$290$	$145$	?	?	?	?	?	Non-Solvable - 2
33553920.a	$\SOMinus(4,16)$	$\POMinus(4,16)$	$2^{9} \cdot 3 \cdot 5 \cdot 17 \cdot 257$	$2^{2} \cdot 3 \cdot 5 \cdot 17 \cdot 257$	$-1$	$1$	$2$	$2$	154				$C_1$	$\SOMinus(4,16)$	$\SL(2,256)$	$C_{2}$	$2^{11} \cdot 3 \cdot 5 \cdot 17 \cdot 257$	$2^{2}$		$257$	$256$	$256$	$256$	?	?	?	Non-Solvable - 2
47045520.c	$\POMinus(4,19)$	$\POMinus(4,19)$	$2^{4} \cdot 3^{2} \cdot 5 \cdot 19^{2} \cdot 181$	$2^{2} \cdot 3^{2} \cdot 5 \cdot 19 \cdot 181$	$-1$	$1$	$2$	$2$	123				$C_1$	$\POMinus(4,19)$	$\PSL(2,361)$	$C_{2}$	$2^{5} \cdot 3^{2} \cdot 5 \cdot 19^{2} \cdot 181$	$2$		$362$	$181$	?	?	?	?	?	Non-Solvable - 2
148035360.c	$\POMinus(4,23)$	$\POMinus(4,23)$	$2^{5} \cdot 3 \cdot 5 \cdot 11 \cdot 23^{2} \cdot 53$	$2^{3} \cdot 3 \cdot 5 \cdot 11 \cdot 23 \cdot 53$	$-1$	$1$	$2$	$2$	171				$C_1$	$\POMinus(4,23)$	$\PSL(2,529)$	$C_{2}$	$2^{6} \cdot 3 \cdot 5 \cdot 11 \cdot 23^{2} \cdot 53$	$2$		$530$	$265$	?	?	?	?	?	Non-Solvable - 2
394813440.a	$\SOMinus(8,2)$	$\POMinus(8,2)$	$2^{13} \cdot 3^{4} \cdot 5 \cdot 7 \cdot 17$	$2^{3} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 17$	$-1$	$1$	$2$	$2$	60				$C_1$	$\SOMinus(8,2)$	$\OmegaMinus(8,2)$	$C_{2}$	$2^{13} \cdot 3^{4} \cdot 5 \cdot 7 \cdot 17$	$1$		$119$	$34$	$34$	$34$	?	?	?	Non-Solvable - 2
2036736000.a	$\SOMinus(6,4)$	$\POMinus(6,4)$	$2^{13} \cdot 3^{2} \cdot 5^{3} \cdot 13 \cdot 17$	$2^{3} \cdot 3 \cdot 5 \cdot 13 \cdot 17$	$-1$	$1$	$2$	$2$	86				$C_1$	$\SOMinus(6,4)$	$\SU(4,4)$	$C_{2}$	$2^{14} \cdot 3^{2} \cdot 5^{3} \cdot 13 \cdot 17$	$2$		$325$	$52$	$52$	$52$	?	?	?	Non-Solvable - 2
29484000000.c	$\POMinus(6,5)$	$\POMinus(6,5)$	$2^{8} \cdot 3^{4} \cdot 5^{6} \cdot 7 \cdot 13$	$2^{3} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 13$	$-1$	$1$	$2$	$2$	116			$3$	$C_1$	$\POMinus(6,5)$	$\PSU(4,5)$	$C_{2}$	$2^{9} \cdot 3^{4} \cdot 5^{6} \cdot 7 \cdot 13$	$2$	$756$	$756$	$105$	?	?	?	?	?	Non-Solvable - 2
4662288691200.c	$\POMinus(6,7)$	$\POMinus(6,7)$	$2^{12} \cdot 3^{2} \cdot 5^{2} \cdot 7^{6} \cdot 43$	$2^{4} \cdot 3 \cdot 5^{2} \cdot 7 \cdot 43$	$-1$	$1$	$3$	$2$	239			$6$	$C_1$	$\POMinus(6,7)$	$\PSU(4,7)$	$C_{2}^{2}$	$2^{13} \cdot 3^{2} \cdot 5^{2} \cdot 7^{6} \cdot 43$	$2$			$301$	?	?	?	?	?	Non-Solvable - 3
20303937239040.c	$\POMinus(8,3)$	$\POMinus(8,3)$	$2^{11} \cdot 3^{12} \cdot 5 \cdot 7 \cdot 13 \cdot 41$	$2^{3} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 13 \cdot 41$	$-1$	$1$	$2$	$2$	188			$3$	$C_1$	$\POMinus(8,3)$	$\OmegaMinus(8,3)$	$C_{2}$	$2^{12} \cdot 3^{12} \cdot 5 \cdot 7 \cdot 13 \cdot 41$	$2$	$1066$	$1066$	$246$	?	?	?	?	?	Non-Solvable - 2
69387579555840.a	$\SOMinus(6,8)$	$\POMinus(6,8)$	$2^{19} \cdot 3^{7} \cdot 5 \cdot 7^{2} \cdot 13 \cdot 19$	$2^{3} \cdot 3^{3} \cdot 5 \cdot 7 \cdot 13 \cdot 19$	$-1$	$1$	$2$	$2$	424				$C_1$	$\SOMinus(6,8)$	$\SU(4,8)$	$C_{2}$	$2^{19} \cdot 3^{8} \cdot 5 \cdot 7^{2} \cdot 13 \cdot 19$	$3$		$4617$	$456$	?	?	?	?	?	Non-Solvable - 2

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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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