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

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Parity Cyclic Solvable Primitive Equivalent siblings
Degree $T$-number Order Group Nilpotency class
$\#\Aut(F/K)$ Transitivity
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Results (34 matches)

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Label Name Order Parity Solvable $\#\Aut(F/K)$ Subfields Low Degree Siblings
9T1 $C_9$ $9$ $1$ ✓ $9$ $C_3$
9T2 $C_3^2$ $9$ $1$ ✓ $9$ $C_3$ x 4
9T3 $D_{9}$ $18$ $1$ ✓ $1$ $S_3$ 18T5
9T4 $S_3\times C_3$ $18$ $-1$ ✓ $3$ $C_3$, $S_3$ 6T5, 18T3
9T5 $C_3^2:C_2$ $18$ $1$ ✓ $1$ $S_3$ x 4 18T4
9T6 $C_9:C_3$ $27$ $1$ ✓ $3$ $C_3$ 27T5
9T7 $C_3^2:C_3$ $27$ $1$ ✓ $3$ $C_3$ 9T7 x 3, 27T3
9T8 $S_3^2$ $36$ $-1$ ✓ $1$ $S_3$ x 2 6T9, 12T16, 18T9, 18T11 x 2, 36T13
9T9 $C_3^2:C_4$ $36$ $1$ ✓ $1$ 6T10 x 2, 12T17 x 2, 18T10, 36T14
9T10 $(C_9:C_3):C_2$ $54$ $1$ ✓ $1$ $S_3$ 18T18, 27T14
9T11 $C_3^2 : C_6$ $54$ $1$ ✓ $1$ $S_3$ 9T13, 18T20, 18T21, 18T22, 27T11
9T12 $(C_3^2:C_3):C_2$ $54$ $-1$ ✓ $3$ $S_3$ 9T12 x 3, 18T24 x 4, 27T6
9T13 $C_3^2 : S_3 $ $54$ $-1$ ✓ $1$ $C_3$ 9T11, 18T20, 18T21, 18T22, 27T11
9T14 $C_3^2:Q_8$ $72$ $1$ ✓ $1$ 12T47, 18T35 x 3, 24T82, 36T55
9T15 $C_3^2:C_8$ $72$ $-1$ ✓ $1$ 12T46, 18T28, 24T81, 36T49
9T16 $S_3^2:C_2$ $72$ $-1$ ✓ $1$ 6T13 x 2, 12T34 x 2, 12T35 x 2, 12T36 x 2, 18T34 x 2, 18T36, 24T72 x 2, 36T53, 36T54 x 2
9T17 $C_3 \wr C_3 $ $81$ $1$ ✓ $3$ $C_3$ 9T17 x 2, 27T19, 27T21, 27T27 x 3
9T18 $C_3^2 : D_{6} $ $108$ $-1$ ✓ $1$ $S_3$ 9T18, 18T51 x 2, 18T55 x 2, 18T56, 18T57 x 2, 27T29, 36T87 x 2, 36T90
9T19 $(C_3^2:C_8):C_2$ $144$ $-1$ ✓ $1$ 12T84, 18T68, 18T71, 18T73, 24T278, 24T279, 24T280, 36T171, 36T172, 36T175
9T20 $C_3 \wr S_3 $ $162$ $-1$ ✓ $3$ $S_3$ 9T20 x 2, 18T86 x 3, 27T37, 27T50 x 3, 27T70
9T21 $(C_3^3:C_3):C_2$ $162$ $1$ ✓ $1$ $S_3$ 9T21 x 2, 18T88 x 3, 27T51 x 3, 27T52, 27T67
9T22 $(C_3^3:C_3):C_2$ $162$ $-1$ ✓ $1$ $C_3$ 9T22 x 2, 18T85 x 3, 27T53 x 3, 27T62, 27T63
9T23 $(C_3^2:Q_8):C_3$ $216$ $1$ ✓ $1$ 12T122, 24T562, 24T569, 27T82, 36T287, 36T309
9T24 $((C_3^3:C_3):C_2):C_2$ $324$ $-1$ ✓ $1$ $S_3$ 9T24 x 2, 18T129 x 3, 18T136 x 3, 18T137 x 3, 27T121, 27T128 x 3, 27T129, 36T502 x 3
9T25 $((C_3 \times (C_3^2 : C_2)) : C_2) : C_3$ $324$ $1$ ✓ $1$ $C_3$ 12T132 x 2, 12T133, 18T141 x 2, 18T142, 18T143, 27T130, 27T131, 36T511, 36T512, 36T524, 36T546 x 2, 36T547
9T26 $((C_3^2:Q_8):C_3):C_2$ $432$ $-1$ ✓ $1$ 12T157, 18T157, 24T1325, 24T1326, 24T1327, 24T1334, 27T139, 36T689, 36T709
9T27 $\PSL(2,8)$ $504$ $1$ $1$ 28T70, 36T712
9T28 $S_3 \wr C_3 $ $648$ $-1$ ✓ $1$ $C_3$ 12T176, 18T197 x 2, 18T198 x 2, 18T202, 18T204, 18T206, 18T207, 24T1528, 24T1539, 27T210, 27T213, 36T1094 x 2, 36T1095, 36T1096 x 2, 36T1097, 36T1099, 36T1101, 36T1102, 36T1103, 36T1137, 36T1238
9T29 $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ $648$ $-1$ ✓ $1$ $S_3$ 12T175, 18T219, 18T220, 18T223, 18T224, 24T1527, 24T1540, 27T214, 27T217, 36T1126, 36T1127, 36T1128, 36T1129, 36T1131, 36T1132, 36T1139, 36T1237
9T30 $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ $648$ $1$ ✓ $1$ $S_3$ 12T177 x 2, 12T178, 18T217, 18T218, 18T221, 18T222, 24T1529 x 2, 24T1530, 27T211, 27T216, 36T1121, 36T1122, 36T1123, 36T1124, 36T1125, 36T1130, 36T1140, 36T1239 x 2, 36T1240
9T31 $S_3\wr S_3$ $1296$ $-1$ ✓ $1$ $S_3$ 12T213, 18T300, 18T303, 18T311, 18T312, 18T314, 18T315, 18T319, 18T320, 24T2893, 24T2894, 24T2895, 24T2912, 27T296, 27T298, 36T2197, 36T2198, 36T2199, 36T2201, 36T2202, 36T2210, 36T2211, 36T2212, 36T2213, 36T2214, 36T2215, 36T2216, 36T2217, 36T2218, 36T2219, 36T2220, 36T2225, 36T2226, 36T2229, 36T2305
9T32 $\mathrm{P}\Gamma\mathrm{L}(2,8)$ $1512$ $1$ $1$ 27T391, 28T165, 36T2342
9T33 $A_9$ $181440$ $1$ $1$ 36T23796
9T34 $S_9$ $362880$ $-1$ $1$ 18T887, 36T28590
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