Galois group search results

Results (1-50 of 512614 matches)

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Label	Name	Order	Parity	Solvable	Nil. class	$\#\Aut(F/K)$	Conj. classes	Subfields	Low Degree Siblings
1T1	Trivial	$1$	$1$	✓	$0$	$1$	$1$
2T1	$C_2$	$2$	$-1$	✓	$1$	$2$	$2$
3T1	$C_3$	$3$	$1$	✓	$1$	$3$	$3$
3T2	$S_3$	$6$	$-1$	✓	$-1$	$1$	$3$		6T2
4T1	$C_4$	$4$	$-1$	✓	$1$	$4$	$4$	$C_2$
4T2	$C_2^2$	$4$	$1$	✓	$1$	$4$	$4$	$C_2$ x 3
4T3	$D_{4}$	$8$	$-1$	✓	$2$	$2$	$5$	$C_2$	4T3, 8T4
4T4	$A_4$	$12$	$1$	✓	$-1$	$1$	$4$		6T4, 12T4
4T5	$S_4$	$24$	$-1$	✓	$-1$	$1$	$5$		6T7, 6T8, 8T14, 12T8, 12T9, 24T10
5T1	$C_5$	$5$	$1$	✓	$1$	$5$	$5$
5T2	$D_{5}$	$10$	$1$	✓	$-1$	$1$	$4$		10T2
5T3	$F_5$	$20$	$-1$	✓	$-1$	$1$	$5$		10T4, 20T5
5T4	$A_5$	$60$	$1$		$-1$	$1$	$5$		6T12, 10T7, 12T33, 15T5, 20T15, 30T9
5T5	$S_5$	$120$	$-1$		$-1$	$1$	$7$		6T14, 10T12, 10T13, 12T74, 15T10, 20T30, 20T32, 20T35, 24T202, 30T22, 30T25, 30T27, 40T62
6T1	$C_6$	$6$	$-1$	✓	$1$	$6$	$6$	$C_2$, $C_3$
6T2	$S_3$	$6$	$-1$	✓	$-1$	$6$	$3$	$C_2$, $S_3$	3T2
6T3	$D_{6}$	$12$	$-1$	✓	$-1$	$2$	$6$	$C_2$, $S_3$	6T3, 12T3
6T4	$A_4$	$12$	$1$	✓	$-1$	$2$	$4$	$C_3$	4T4, 12T4
6T5	$S_3\times C_3$	$18$	$-1$	✓	$-1$	$3$	$9$	$C_2$	9T4, 18T3
6T6	$A_4\times C_2$	$24$	$-1$	✓	$-1$	$2$	$8$	$C_3$	8T13, 12T6, 12T7, 24T9
6T7	$S_4$	$24$	$1$	✓	$-1$	$2$	$5$	$S_3$	4T5, 6T8, 8T14, 12T8, 12T9, 24T10
6T8	$S_4$	$24$	$-1$	✓	$-1$	$2$	$5$	$S_3$	4T5, 6T7, 8T14, 12T8, 12T9, 24T10
6T9	$S_3^2$	$36$	$-1$	✓	$-1$	$1$	$9$	$C_2$	9T8, 12T16, 18T9, 18T11 x 2, 36T13
6T10	$C_3^2:C_4$	$36$	$1$	✓	$-1$	$1$	$6$	$C_2$	6T10, 9T9, 12T17 x 2, 18T10, 36T14
6T11	$S_4\times C_2$	$48$	$-1$	✓	$-1$	$2$	$10$	$S_3$	6T11, 8T24 x 2, 12T21, 12T22, 12T23 x 2, 12T24 x 2, 16T61, 24T46, 24T47, 24T48 x 2
6T12	$\PSL(2,5)$	$60$	$1$		$-1$	$1$	$5$		5T4, 10T7, 12T33, 15T5, 20T15, 30T9
6T13	$C_3^2:D_4$	$72$	$-1$	✓	$-1$	$1$	$9$	$C_2$	6T13, 9T16, 12T34 x 2, 12T35 x 2, 12T36 x 2, 18T34 x 2, 18T36, 24T72 x 2, 36T53, 36T54 x 2
6T14	$\PGL(2,5)$	$120$	$-1$		$-1$	$1$	$7$		5T5, 10T12, 10T13, 12T74, 15T10, 20T30, 20T32, 20T35, 24T202, 30T22, 30T25, 30T27, 40T62
6T15	$A_6$	$360$	$1$		$-1$	$1$	$7$		6T15, 10T26, 15T20 x 2, 20T89, 30T88 x 2, 36T555, 40T304, 45T49
6T16	$S_6$	$720$	$-1$		$-1$	$1$	$11$		6T16, 10T32, 12T183 x 2, 15T28 x 2, 20T145, 20T149 x 2, 30T164 x 2, 30T166 x 2, 30T176 x 2, 36T1252, 40T589, 40T592 x 2, 45T96
7T1	$C_7$	$7$	$1$	✓	$1$	$7$	$7$
7T2	$D_{7}$	$14$	$-1$	✓	$-1$	$1$	$5$		14T2
7T3	$C_7:C_3$	$21$	$1$	✓	$-1$	$1$	$5$		21T2
7T4	$F_7$	$42$	$-1$	✓	$-1$	$1$	$7$		14T4, 21T4, 42T4
7T5	$\GL(3,2)$	$168$	$1$		$-1$	$1$	$6$		7T5, 8T37, 14T10 x 2, 21T14, 24T284, 28T32, 42T37, 42T38 x 2
7T6	$A_7$	$2520$	$1$		$-1$	$1$	$9$		15T47 x 2, 21T33, 35T28, 42T294, 42T299
7T7	$S_7$	$5040$	$-1$		$-1$	$1$	$15$		14T46, 21T38, 30T565, 35T31, 42T411, 42T412, 42T413, 42T418
8T1	$C_8$	$8$	$-1$	✓	$1$	$8$	$8$	$C_2$, $C_4$
8T2	$C_4\times C_2$	$8$	$1$	✓	$1$	$8$	$8$	$C_2$ x 3, $C_4$ x 2, $C_2^2$
8T3	$C_2^3$	$8$	$1$	✓	$1$	$8$	$8$	$C_2$ x 7, $C_2^2$ x 7
8T4	$D_4$	$8$	$1$	✓	$2$	$8$	$5$	$C_2$ x 3, $C_2^2$, $D_{4}$ x 2	4T3 x 2
8T5	$Q_8$	$8$	$1$	✓	$2$	$8$	$5$	$C_2$ x 3, $C_2^2$
8T6	$D_{8}$	$16$	$-1$	✓	$3$	$2$	$7$	$C_2$, $D_{4}$	8T6, 16T13
8T7	$C_8:C_2$	$16$	$-1$	✓	$2$	$4$	$10$	$C_2$, $C_4$	16T6
8T8	$QD_{16}$	$16$	$-1$	✓	$3$	$2$	$7$	$C_2$, $D_{4}$	16T12
8T9	$D_4\times C_2$	$16$	$1$	✓	$2$	$4$	$10$	$C_2$ x 3, $C_2^2$, $D_{4}$ x 2	8T9 x 3, 16T9
8T10	$C_2^2:C_4$	$16$	$1$	✓	$2$	$4$	$10$	$C_2$, $C_4$, $D_{4}$ x 2	8T10, 16T10
8T11	$Q_8:C_2$	$16$	$1$	✓	$2$	$4$	$10$	$C_2$ x 3, $C_2^2$	8T11 x 2, 16T11
8T12	$\SL(2,3)$	$24$	$1$	✓	$-1$	$2$	$7$	$A_4$	24T7
8T13	$A_4\times C_2$	$24$	$1$	✓	$-1$	$2$	$8$	$C_2$, $A_4$	6T6, 12T6, 12T7, 24T9

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