Galois group search results

The results below are complete, since the LMFDB contains all transitive groups of degree at most 47 (except 32)

Results (1-50 of 2392 matches)

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Label	Name	Degree	Order	Parity	Solvable	Nil. class	$\#\Aut(F/K)$	Transitivity	Conj. classes	Subfields	Low Degree Siblings
27T1	$C_{27}$	$27$	$27$	$1$	✓	$1$	$27$	$1$	$27$	$C_3$, $C_9$
27T2	$C_3\times C_9$	$27$	$27$	$1$	✓	$1$	$27$	$1$	$27$	$C_3$ x 4, $C_9$ x 3, $C_3^2$
27T3	$\He_3$	$27$	$27$	$1$	✓	$2$	$27$	$1$	$11$	$C_3$ x 4, $C_3^2$, $C_3^2:C_3$ x 4	9T7 x 4
27T4	$C_3^3$	$27$	$27$	$1$	✓	$1$	$27$	$1$	$27$	$C_3$ x 13, $C_3^2$ x 13
27T5	$C_9:C_3$	$27$	$27$	$1$	✓	$2$	$27$	$1$	$11$	$C_3$ x 4, $C_3^2$, $C_9:C_3$	9T6
27T6	$C_3^2:S_3$	$27$	$54$	$1$	✓	$-1$	$3$	$1$	$10$	$S_3$ x 4, $C_3^2:C_2$, $(C_3^2:C_3):C_2$ x 4	9T12 x 4, 18T24 x 4
27T7	$C_3^2:S_3$	$27$	$54$	$-1$	✓	$-1$	$1$	$1$	$15$	$S_3$ x 13, $C_3^2:C_2$ x 13
27T8	$D_{27}$	$27$	$54$	$-1$	✓	$-1$	$1$	$1$	$15$	$S_3$, $D_{9}$
27T9	$C_3\times D_9$	$27$	$54$	$1$	✓	$-1$	$3$	$1$	$18$	$C_3$, $S_3$, $D_{9}$, $S_3\times C_3$	18T19
27T10	$C_3:D_9$	$27$	$54$	$-1$	✓	$-1$	$1$	$1$	$15$	$S_3$ x 4, $D_{9}$ x 3, $C_3^2:C_2$
27T11	$C_3^2:C_6$	$27$	$54$	$1$	✓	$-1$	$3$	$1$	$10$	$C_3$, $S_3$, $S_3\times C_3$, $C_3^2 : C_6$, $C_3^2 : S_3 $	9T11, 9T13, 18T20, 18T21, 18T22
27T12	$S_3\times C_9$	$27$	$54$	$-1$	✓	$-1$	$9$	$1$	$27$	$C_3$, $S_3$, $C_9$, $S_3\times C_3$	18T16
27T13	$C_3^2:C_6$	$27$	$54$	$1$	✓	$-1$	$3$	$1$	$18$	$C_3$, $S_3$ x 4, $S_3\times C_3$ x 4, $C_3^2:C_2$	18T23 x 4
27T14	$C_9:C_6$	$27$	$54$	$1$	✓	$-1$	$3$	$1$	$10$	$C_3$, $S_3$, $S_3\times C_3$, $(C_9:C_3):C_2$	9T10, 18T18
27T15	$S_3\times C_3^2$	$27$	$54$	$-1$	✓	$-1$	$9$	$1$	$27$	$C_3$ x 4, $S_3$, $C_3^2$, $S_3\times C_3$ x 4	18T17 x 4
27T16	$C_9:C_3^2$	$27$	$81$	$1$	✓	$2$	$9$	$1$	$33$	$C_3$ x 4, $C_3^2$, $C_9:C_3$ x 3	27T16 x 2
27T17	$C_3^2:C_9$	$27$	$81$	$1$	✓	$2$	$9$	$1$	$33$	$C_3$, $C_9$, $C_9:C_3$ x 2, $C_3^2:C_3$	27T17 x 2
27T18	$C_3\times \He_3$	$27$	$81$	$1$	✓	$2$	$9$	$1$	$33$	$C_3$ x 4, $C_3^2$, $C_3^2:C_3$ x 3	27T18 x 11
27T19	$C_3\wr C_3$	$27$	$81$	$1$	✓	$3$	$9$	$1$	$17$	$C_3$ x 4, $C_3^2$, $C_3 \wr C_3 $ x 3	9T17 x 3, 27T21, 27T27 x 3
27T20	$C_3.\He_3$	$27$	$81$	$1$	✓	$3$	$3$	$1$	$17$	$C_3$, $C_3^2:C_3$	27T26
27T21	$C_3\wr C_3$	$27$	$81$	$1$	✓	$3$	$3$	$1$	$17$	$C_3$, $C_3^2:C_3$	9T17 x 3, 27T19, 27T27 x 3
27T22	$C_{27}:C_3$	$27$	$81$	$1$	✓	$2$	$9$	$1$	$33$	$C_3$, $C_9$
27T23	$\He_3:C_3$	$27$	$81$	$1$	✓	$3$	$3$	$1$	$17$	$C_3$, $C_3^2:C_3$	27T23 x 2, 27T24
27T24	$\He_3:C_3$	$27$	$81$	$1$	✓	$3$	$9$	$1$	$17$	$C_3$ x 4, $C_3^2$	27T23 x 3
27T25	$C_3.\He_3$	$27$	$81$	$1$	✓	$3$	$9$	$1$	$17$	$C_3$ x 4, $C_3^2$
27T26	$C_3.\He_3$	$27$	$81$	$1$	✓	$3$	$9$	$1$	$17$	$C_3$ x 4, $C_3^2$	27T20
27T27	$C_3\wr C_3$	$27$	$81$	$1$	✓	$3$	$9$	$1$	$17$	$C_3$, $C_3^2:C_3$, $C_3 \wr C_3 $ x 2	9T17 x 3, 27T19, 27T21, 27T27 x 2
27T28	$C_9.C_3^2$	$27$	$81$	$1$	✓	$2$	$9$	$1$	$33$	$C_3$ x 4, $C_3^2$	27T28 x 3
27T29	$C_3^2:D_6$	$27$	$108$	$1$	✓	$-1$	$1$	$1$	$11$	$S_3$ x 2, $S_3^2$, $C_3^2 : D_{6} $ x 2	9T18 x 2, 18T51 x 2, 18T55 x 2, 18T56, 18T57 x 2, 36T87 x 2, 36T90
27T30	$S_3\times D_9$	$27$	$108$	$-1$	✓	$-1$	$1$	$1$	$18$	$S_3$ x 2, $D_{9}$, $S_3^2$	18T50, 36T86
27T31	$C_3^3:C_4$	$27$	$108$	$-1$	✓	$-1$	$1$	$1$	$12$	$S_3$, $C_3^2:C_4$	12T72 x 2, 18T54 x 2, 36T89 x 2, 36T94 x 2
27T32	$\He_3:C_4$	$27$	$108$	$1$	✓	$-1$	$3$	$1$	$14$	$C_3^2:C_4$	18T49 x 2, 36T85 x 2
27T33	$C_3^2:C_{12}$	$27$	$108$	$1$	✓	$-1$	$3$	$1$	$18$	$C_3$, $C_3^2:C_4$	12T73 x 2, 18T44 x 2, 36T81 x 2, 36T95 x 2
27T34	$C_3:S_3^2$	$27$	$108$	$-1$	✓	$-1$	$1$	$1$	$18$	$S_3$ x 5, $C_3^2:C_2$, $S_3^2$ x 4	18T58 x 4, 36T91 x 4
27T35	$C_3:S_3^2$	$27$	$108$	$1$	✓	$-1$	$1$	$1$	$15$	$S_3$ x 3, $S_3^2$ x 3	12T71, 18T53 x 3, 36T88 x 3, 36T93
27T36	$C_3\times S_3^2$	$27$	$108$	$-1$	✓	$-1$	$3$	$1$	$27$	$C_3$, $S_3$ x 2, $S_3\times C_3$ x 2, $S_3^2$	12T70, 18T43, 18T46 x 2, 36T80, 36T82 x 2, 36T92
27T37	$C_3\wr S_3$	$27$	$162$	$-1$	✓	$-1$	$9$	$1$	$22$	$C_3$, $S_3$, $S_3\times C_3$, $C_3 \wr S_3 $ x 3	9T20 x 3, 18T86 x 3, 27T50 x 3, 27T70
27T38	$\He_3.S_3$	$27$	$162$	$1$	✓	$-1$	$1$	$1$	$13$	$C_3$, $S_3$, $S_3\times C_3$	27T64
27T39	$\He_3.C_6$	$27$	$162$	$-1$	✓	$-1$	$9$	$1$	$30$	$C_3$, $S_3$, $S_3\times C_3$	27T39 x 3
27T40	$\He_3.C_6$	$27$	$162$	$-1$	✓	$-1$	$9$	$1$	$22$	$C_3$, $S_3$, $S_3\times C_3$	27T49
27T41	$\He_3.S_3$	$27$	$162$	$1$	✓	$-1$	$1$	$1$	$13$	$C_3$, $S_3$, $S_3\times C_3$	27T72
27T42	$\He_3.S_3$	$27$	$162$	$-1$	✓	$-1$	$1$	$1$	$13$	$S_3$ x 4, $C_3^2:C_2$	27T68
27T43	$(C_3\times C_9).S_3$	$27$	$162$	$-1$	✓	$-1$	$1$	$1$	$13$	$S_3$ x 4, $C_3^2:C_2$
27T44	$\He_3:S_3$	$27$	$162$	$-1$	✓	$-1$	$1$	$1$	$13$	$S_3$ x 4, $C_3^2:C_2$	27T66 x 3
27T45	$\He_3.C_6$	$27$	$162$	$-1$	✓	$-1$	$9$	$1$	$22$	$C_3$, $S_3$, $S_3\times C_3$	27T69
27T46	$C_3^3:S_3$	$27$	$162$	$-1$	✓	$-1$	$9$	$1$	$30$	$C_3$, $S_3$, $S_3\times C_3$, $(C_3^2:C_3):C_2$ x 3	27T46 x 3
27T47	$C_3^2:C_{18}$	$27$	$162$	$-1$	✓	$-1$	$3$	$1$	$30$	$C_3$, $C_9$, $C_3^2 : S_3 $	18T82
27T48	$C_3^3:C_6$	$27$	$162$	$-1$	✓	$-1$	$3$	$1$	$30$	$C_3$ x 4, $C_3^2$, $C_3^2 : S_3 $	18T76, 18T78, 18T81 x 2, 27T60 x 3
27T49	$\He_3.C_6$	$27$	$162$	$-1$	✓	$-1$	$3$	$1$	$22$	$C_3$, $C_3^2 : S_3 $	27T40
27T50	$C_3\wr S_3$	$27$	$162$	$1$	✓	$-1$	$3$	$1$	$22$	$S_3$, $C_3^2 : C_6$, $C_3 \wr S_3 $	9T20 x 3, 18T86 x 3, 27T37, 27T50 x 2, 27T70

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