| 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 |
| 6.1 |
$S_3$ |
$\GOrth(3,2)$ |
$2 \cdot 3$ |
$2 \cdot 3$ |
$-1$ |
$2$ |
$2$ |
$2$ |
3 |
$6$ |
$4$ |
$3$ |
$C_1$ |
$S_3$ |
$C_3$ |
$C_{2}$ |
$2 \cdot 3$ |
$1$ |
$3$ |
$3$ |
$2$ |
$2$ |
$2$ |
$2$ |
$2$ |
$2$ |
Solvable - 2 |
| 48.48 |
$C_2\times S_4$ |
$\GOrth(3,3)$ |
$2^{4} \cdot 3$ |
$2^{2} \cdot 3$ |
$-1$ |
$3$ |
$5$ |
$2$ |
10 |
$98$ |
$33$ |
$9$ |
$C_2$ |
$S_4$ |
$A_4$ |
$C_{2}^{2}$ |
$2^{4} \cdot 3$ |
$2$ |
$6$ |
$6$ |
$3$ |
$3$ |
$3$ |
$3$ |
$3$ |
$3$ |
Solvable - 3 |
| 180.19 |
$\GL(2,4)$ |
$\GOrth(3,4)$ |
$2^{2} \cdot 3^{2} \cdot 5$ |
$2 \cdot 3 \cdot 5$ |
$-1$ |
$1$ |
$2$ |
$2$ |
15 |
$148$ |
$21$ |
$4$ |
$C_3$ |
$A_5$ |
$A_5$ |
$C_{3}$ |
$2^{4} \cdot 3 \cdot 5$ |
$2^{2}$ |
$15$ |
$8$ |
$3$ |
$6$ |
$8$ |
$3$ |
$5$ |
$6$ |
Non-Solvable - 2 |
| 480.943 |
$C_4\times S_5$ |
$\GOrth(3,5)$ |
$2^{5} \cdot 3 \cdot 5$ |
$2^{2} \cdot 3 \cdot 5$ |
$-1$ |
$1$ |
$4$ |
$2$ |
28 |
$956$ |
$98$ |
$11$ |
$C_4$ |
$S_5$ |
$A_5$ |
$C_{2} \times C_{4}$ |
$2^{5} \cdot 3 \cdot 5$ |
$2^{2}$ |
$20$ |
$9$ |
$4$ |
$8$ |
$8$ |
$4$ |
$6$ |
$6$ |
Non-Solvable - 4 |
| 720.763 |
$S_6$ |
$\GOrth(5,2)$ |
$2^{4} \cdot 3^{2} \cdot 5$ |
$2^{2} \cdot 3 \cdot 5$ |
$-1$ |
$1$ |
$2$ |
$2$ |
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 |
| 2016.b |
$C_6\times \PGL(2,7)$ |
$\GOrth(3,7)$ |
$2^{5} \cdot 3^{2} \cdot 7$ |
$2^{3} \cdot 3 \cdot 7$ |
$-1$ |
$1$ |
$4$ |
$2$ |
54 |
$3362$ |
$157$ |
$14$ |
$C_6$ |
$\PGL(2,7)$ |
$\PSL(2,7)$ |
$C_{2} \times C_{6}$ |
$2^{6} \cdot 3 \cdot 7$ |
$2^{2}$ |
$48$ |
$13$ |
$6$ |
$12$ |
$12$ |
$6$ |
$8$ |
$8$ |
Non-Solvable - 4 |
| 3528.a |
$\GL(2,8)$ |
$\GOrth(3,8)$ |
$2^{3} \cdot 3^{2} \cdot 7^{2}$ |
$2 \cdot 3^{2} \cdot 7$ |
$-1$ |
$1$ |
$2$ |
$2$ |
63 |
|
|
|
$C_7$ |
$\SL(2,8)$ |
$\SL(2,8)$ |
$C_{7}$ |
$2^{4} \cdot 3^{4} \cdot 7$ |
$2 \cdot 3^{2}$ |
|
$16$ |
$7$ |
$14$ |
$42$ |
? |
? |
? |
Non-Solvable - 2 |
| 5760.b |
$C_8\times \PGL(2,9)$ |
$\GOrth(3,9)$ |
$2^{7} \cdot 3^{2} \cdot 5$ |
$2^{3} \cdot 3 \cdot 5$ |
$-1$ |
$1$ |
$5$ |
$2$ |
88 |
$8183$ |
$200$ |
$15$ |
$C_8$ |
$\PGL(2,9)$ |
$A_6$ |
$C_{2} \times C_{8}$ |
$2^{8} \cdot 3^{2} \cdot 5$ |
$2^{4}$ |
$80$ |
$18$ |
$8$ |
$16$ |
$36$ |
$8$ |
$10$ |
$13$ |
Non-Solvable - 5 |
| 13200.b |
$C_{10}\times \PGL(2,11)$ |
$\GOrth(3,11)$ |
$2^{4} \cdot 3 \cdot 5^{2} \cdot 11$ |
$2^{2} \cdot 3 \cdot 5 \cdot 11$ |
$-1$ |
$1$ |
$4$ |
$2$ |
130 |
$13320$ |
$206$ |
$14$ |
$C_{10}$ |
$\PGL(2,11)$ |
$\PSL(2,11)$ |
$C_{2} \times C_{10}$ |
$2^{6} \cdot 3 \cdot 5 \cdot 11$ |
$2^{3}$ |
$120$ |
$19$ |
$10$ |
$20$ |
$40$ |
$10$ |
$12$ |
$14$ |
Non-Solvable - 4 |
| 26208.b |
$C_{12}\times \PGL(2,13)$ |
$\GOrth(3,13)$ |
$2^{5} \cdot 3^{2} \cdot 7 \cdot 13$ |
$2^{2} \cdot 3 \cdot 7 \cdot 13$ |
$-1$ |
$1$ |
$5$ |
$2$ |
180 |
$35940$ |
$368$ |
$22$ |
$C_{12}$ |
$\PGL(2,13)$ |
$\PSL(2,13)$ |
$C_{2} \times C_{12}$ |
$2^{6} \cdot 3 \cdot 7 \cdot 13$ |
$2^{3}$ |
$168$ |
$21$ |
$12$ |
$24$ |
$52$ |
$12$ |
$14$ |
$17$ |
Non-Solvable - 5 |
| 61200.a |
$\GL(2,16)$ |
$\GOrth(3,16)$ |
$2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 17$ |
$2 \cdot 3 \cdot 5 \cdot 17$ |
$-1$ |
$1$ |
$3$ |
$2$ |
255 |
|
|
|
$C_{15}$ |
$\SL(2,16)$ |
$\SL(2,16)$ |
$C_{15}$ |
$2^{9} \cdot 3 \cdot 5 \cdot 17$ |
$2^{5}$ |
|
$25$ |
$15$ |
$30$ |
$128$ |
? |
? |
? |
Non-Solvable - 3 |
| 78336.b |
$C_{16}\times \PGL(2,17)$ |
$\GOrth(3,17)$ |
$2^{9} \cdot 3^{2} \cdot 17$ |
$2^{4} \cdot 3^{2} \cdot 17$ |
$-1$ |
$1$ |
$6$ |
$2$ |
304 |
$72454$ |
$371$ |
$19$ |
$C_{16}$ |
$\PGL(2,17)$ |
$\PSL(2,17)$ |
$C_{2} \times C_{16}$ |
$2^{9} \cdot 3^{2} \cdot 17$ |
$2^{4}$ |
$288$ |
$34$ |
$16$ |
$32$ |
$128$ |
$16$ |
$18$ |
$24$ |
Non-Solvable - 6 |
| 103680.a |
$C_2\times \SO(5,3)$ |
$\GOrth(5,3)$ |
$2^{8} \cdot 3^{4} \cdot 5$ |
$2^{3} \cdot 3^{2} \cdot 5$ |
$-1$ |
$1$ |
$3$ |
$2$ |
50 |
$1046205$ |
$1503$ |
$7$ |
$C_2$ |
$\SO(5,3)$ |
$\SU(4,2)$ |
$C_{2}^{2}$ |
$2^{8} \cdot 3^{4} \cdot 5$ |
$2$ |
$54$ |
$29$ |
$6$ |
$6$ |
$6$ |
? |
? |
? |
Non-Solvable - 3 |
| 123120.b |
$C_{18}\times \PGL(2,19)$ |
$\GOrth(3,19)$ |
$2^{4} \cdot 3^{4} \cdot 5 \cdot 19$ |
$2^{2} \cdot 3^{2} \cdot 5 \cdot 19$ |
$-1$ |
$1$ |
$5$ |
$2$ |
378 |
$106308$ |
$436$ |
$21$ |
$C_{18}$ |
$\PGL(2,19)$ |
$\PSL(2,19)$ |
$C_{2} \times C_{18}$ |
$2^{5} \cdot 3^{3} \cdot 5 \cdot 19$ |
$2^{2} \cdot 3$ |
$360$ |
$31$ |
$18$ |
$36$ |
$108$ |
$18$ |
$20$ |
$24$ |
Non-Solvable - 5 |
| 267168.b |
$C_{22}\times \PGL(2,23)$ |
$\GOrth(3,23)$ |
$2^{5} \cdot 3 \cdot 11^{2} \cdot 23$ |
$2^{3} \cdot 3 \cdot 11 \cdot 23$ |
$-1$ |
$1$ |
$4$ |
$2$ |
550 |
$114026$ |
$291$ |
$14$ |
$C_{22}$ |
$\PGL(2,23)$ |
$\PSL(2,23)$ |
$C_{2} \times C_{22}$ |
$2^{6} \cdot 3 \cdot 5 \cdot 11 \cdot 23$ |
$2^{2} \cdot 5$ |
$528$ |
$37$ |
$22$ |
? |
$220$ |
? |
? |
? |
Non-Solvable - 4 |
| 374400.b |
$\GOrth(3,25)$ |
$\GOrth(3,25)$ |
$2^{7} \cdot 3^{2} \cdot 5^{2} \cdot 13$ |
$2^{3} \cdot 3 \cdot 5 \cdot 13$ |
$-1$ |
$1$ |
$6$ |
$2$ |
648 |
$393240$ |
$821$ |
$30$ |
$C_{24}$ |
$\PGL(2,25)$ |
$\PSL(2,25)$ |
$C_{2} \times C_{24}$ |
$2^{9} \cdot 3 \cdot 5^{2} \cdot 13$ |
$2^{5}$ |
$624$ |
$37$ |
$24$ |
? |
$200$ |
? |
? |
? |
Non-Solvable - 6 |
| 511056.b |
$C_{26}\times \PGL(2,27)$ |
$\GOrth(3,27)$ |
$2^{4} \cdot 3^{3} \cdot 7 \cdot 13^{2}$ |
$2^{2} \cdot 3 \cdot 7 \cdot 13$ |
$-1$ |
$1$ |
$4$ |
$2$ |
754 |
$162720$ |
$282$ |
$14$ |
$C_{26}$ |
$\PGL(2,27)$ |
$\PSL(2,27)$ |
$C_{2} \times C_{26}$ |
$2^{6} \cdot 3^{5} \cdot 7 \cdot 13$ |
$2^{3} \cdot 3^{2}$ |
$728$ |
$43$ |
$26$ |
? |
$312$ |
? |
? |
? |
Non-Solvable - 4 |
| 682080.b |
$\GOrth(3,29)$ |
$\GOrth(3,29)$ |
$2^{5} \cdot 3 \cdot 5 \cdot 7^{2} \cdot 29$ |
$2^{2} \cdot 3 \cdot 5 \cdot 7 \cdot 29$ |
$-1$ |
$1$ |
$5$ |
$2$ |
868 |
$417018$ |
$569$ |
$22$ |
$C_{28}$ |
$\PGL(2,29)$ |
$\PSL(2,29)$ |
$C_{2} \times C_{28}$ |
$2^{6} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 29$ |
$2^{3} \cdot 3$ |
$840$ |
$41$ |
$28$ |
? |
$336$ |
? |
? |
? |
Non-Solvable - 5 |
| 892800.b |
$C_{30}\times \PGL(2,31)$ |
$\GOrth(3,31)$ |
$2^{7} \cdot 3^{2} \cdot 5^{2} \cdot 31$ |
$2^{5} \cdot 3 \cdot 5 \cdot 31$ |
$-1$ |
$1$ |
$5$ |
$2$ |
990 |
$624484$ |
$788$ |
$28$ |
$C_{30}$ |
$\PGL(2,31)$ |
$\PSL(2,31)$ |
$C_{2} \times C_{30}$ |
$2^{10} \cdot 3 \cdot 5 \cdot 31$ |
$2^{4}$ |
$960$ |
$42$ |
$30$ |
? |
$240$ |
? |
? |
? |
Non-Solvable - 5 |
| 1014816.a |
$\GL(2,32)$ |
$\GOrth(3,32)$ |
$2^{5} \cdot 3 \cdot 11 \cdot 31^{2}$ |
$2 \cdot 3 \cdot 11 \cdot 31$ |
$-1$ |
$1$ |
$2$ |
$2$ |
1023 |
|
|
|
$C_{31}$ |
$\SL(2,32)$ |
$\SL(2,32)$ |
$C_{31}$ |
$2^{6} \cdot 3^{2} \cdot 5^{2} \cdot 11 \cdot 31$ |
$2 \cdot 3 \cdot 5^{2}$ |
|
$64$ |
$31$ |
? |
? |
? |
? |
? |
Non-Solvable - 2 |
| 2937600.a |
$\GSp(4,4)$ |
$\GOrth(5,4)$ |
$2^{8} \cdot 3^{3} \cdot 5^{2} \cdot 17$ |
$2^{2} \cdot 3 \cdot 5 \cdot 17$ |
$-1$ |
$1$ |
$2$ |
$2$ |
81 |
|
|
|
$C_3$ |
$\Sp(4,4)$ |
$\Sp(4,4)$ |
$C_{3}$ |
$2^{11} \cdot 3^{2} \cdot 5^{2} \cdot 17$ |
$2^{3}$ |
|
$88$ |
$18$ |
? |
? |
? |
? |
? |
Non-Solvable - 2 |
| 37440000.b |
$\GOrth(5,5)$ |
$\GOrth(5,5)$ |
$2^{9} \cdot 3^{2} \cdot 5^{4} \cdot 13$ |
$2^{3} \cdot 3 \cdot 5 \cdot 13$ |
$-1$ |
$1$ |
$4$ |
$2$ |
188 |
|
|
|
$C_4$ |
$\SO(5,5)$ |
$\PSp(4,5)$ |
$C_{2} \times C_{4}$ |
$2^{9} \cdot 3^{2} \cdot 5^{4} \cdot 13$ |
$2^{2}$ |
|
$160$ |
$26$ |
$52$ |
$52$ |
? |
? |
? |
Non-Solvable - 4 |
| 1659571200.b |
$\GOrth(5,7)$ |
$\GOrth(5,7)$ |
$2^{10} \cdot 3^{3} \cdot 5^{2} \cdot 7^{4}$ |
$2^{4} \cdot 3 \cdot 5^{2} \cdot 7$ |
$-1$ |
$1$ |
$4$ |
$2$ |
462 |
|
|
|
$C_6$ |
$\SO(5,7)$ |
$\PSp(4,7)$ |
$C_{2} \times C_{6}$ |
$2^{11} \cdot 3^{2} \cdot 5^{2} \cdot 7^{4}$ |
$2^{2}$ |
|
$405$ |
$50$ |
? |
? |
? |
? |
? |
Non-Solvable - 4 |
| 7396945920.a |
$\GSp(4,8)$ |
$\GOrth(5,8)$ |
$2^{12} \cdot 3^{4} \cdot 5 \cdot 7^{3} \cdot 13$ |
$2^{2} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 13$ |
$-1$ |
$1$ |
$2$ |
$2$ |
581 |
|
|
|
$C_7$ |
$\Sp(4,8)$ |
$\Sp(4,8)$ |
$C_{7}$ |
$2^{14} \cdot 3^{6} \cdot 5 \cdot 7^{2} \cdot 13$ |
$2^{2} \cdot 3^{2}$ |
|
$592$ |
$196$ |
? |
? |
? |
? |
? |
Non-Solvable - 2 |
| 27545702400.b |
$\GOrth(5,9)$ |
$\GOrth(5,9)$ |
$2^{12} \cdot 3^{8} \cdot 5^{2} \cdot 41$ |
$2^{4} \cdot 3^{2} \cdot 5 \cdot 41$ |
$-1$ |
$1$ |
$5$ |
$2$ |
920 |
|
|
|
$C_8$ |
$\SO(5,9)$ |
$\PSp(4,9)$ |
$C_{2} \times C_{8}$ |
$2^{13} \cdot 3^{8} \cdot 5^{2} \cdot 41$ |
$2^{4}$ |
|
$828$ |
$82$ |
? |
? |
? |
? |
? |
Non-Solvable - 5 |
| 257213088000.b |
$\GOrth(5,11)$ |
$\GOrth(5,11)$ |
$2^{8} \cdot 3^{2} \cdot 5^{3} \cdot 11^{4} \cdot 61$ |
$2^{3} \cdot 3 \cdot 5 \cdot 11 \cdot 61$ |
$-1$ |
$1$ |
$4$ |
$2$ |
1610 |
|
|
|
$C_{10}$ |
|
|
$C_{2} \times C_{10}$ |
$2^{10} \cdot 3^{2} \cdot 5^{2} \cdot 11^{4} \cdot 61$ |
$2^{3}$ |
|
$1471$ |
? |
? |
? |
? |
? |
? |
Non-Solvable - 4 |
| 1644455554560.b |
$\GOrth(5,13)$ |
$\GOrth(5,13)$ |
$2^{9} \cdot 3^{3} \cdot 5 \cdot 7^{2} \cdot 13^{4} \cdot 17$ |
$2^{3} \cdot 3 \cdot 5 \cdot 7 \cdot 13 \cdot 17$ |
$-1$ |
$1$ |
$5$ |
$2$ |
2580 |
|
|
|
$C_{12}$ |
|
|
$C_{2} \times C_{12}$ |
$2^{10} \cdot 3^{2} \cdot 5 \cdot 7^{2} \cdot 13^{4} \cdot 17$ |
$2^{3}$ |
|
$2387$ |
? |
? |
? |
? |
? |
? |
Non-Solvable - 5 |
| 16427999232000.a |
$\GSp(4,16)$ |
$\GOrth(5,16)$ |
$2^{16} \cdot 3^{3} \cdot 5^{3} \cdot 17^{2} \cdot 257$ |
$2^{2} \cdot 3 \cdot 5 \cdot 17 \cdot 257$ |
$-1$ |
$1$ |
$3$ |
$2$ |
4365 |
|
|
|
$C_{15}$ |
|
|
$C_{15}$ |
$2^{22} \cdot 3^{2} \cdot 5^{2} \cdot 17^{2} \cdot 257$ |
$2^{6}$ |
|
$4377$ |
? |
? |
? |
? |
? |
? |
Non-Solvable - 3 |
