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Transitive Groups of degree 9

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Transitive Groups of degree 9
G Name Δ |G| |G| fact. |Z(G)| Properties of G # fields
9T1 C(9)=9 9
32
9 cyclic, semiabelian, even 36
9T2 E(9)=3[x]3 9
32
9 abelian, semiabelian, even 57
9T3 D(9)=9:2 18
2 · 32
1 solvable, semiabelian, even 86
9T4 S(3)[x]3 18
2 · 32
3 solvable, semiabelian 414
9T5 S(3)[1/2]S(3)=32:2 18
2 · 32
1 solvable, semiabelian, even 220
9T6 1/3[33]3 27
33
3 nilpotent, semiabelian, even 54
9T7 E(9):3=[32]3 27
33
3 nilpotent, semiabelian, even 83
9T8 S(3)[x]S(3)=E(9):D4 36
22 · 32
1 solvable, semiabelian 564
9T9 E(9):4 36
22 · 32
1 solvable, primitive, semiabelian, even 113
9T10 [32]S(3)6 54
2 · 33
1 solvable, semiabelian, even 140
9T11 E(9):6=1/2[32:2]S(3) 54
2 · 33
1 solvable, semiabelian, even 144
9T12 [32]S(3) 54
2 · 33
3 solvable, semiabelian 180
9T13 E(9):D6=[32:2]3=[1/2.S(3)2]3 54
2 · 33
1 solvable, semiabelian 156
9T14 M(9)=E(9):Q8 72
23 · 32
1 solvable, primitive, semiabelian, even 103
9T15 E(9):8 72
23 · 32
1 solvable, primitive, semiabelian 81
9T16 E(9):D8 72
23 · 32
1 solvable, primitive, semiabelian 222
9T17 [33]3=3wr3 81
34
3 nilpotent, semiabelian, even 58
9T18 E(9):D12=[32:2]S(3)=[1/2.S(3)2]S(3) 108
22 · 33
1 solvable, semiabelian 431
9T19 E(9):2D8 144
24 · 32
1 solvable, primitive, semiabelian 179
9T20 [33]S(3)=3wrS(3) 162
2 · 34
3 solvable, semiabelian 195
9T21 1/2.[33:2]S(3) 162
2 · 34
1 solvable, semiabelian, even 262
9T22 [33:2]3 162
2 · 34
1 solvable, semiabelian 180
9T23 E(9):2A4 216
23 · 33
1 solvable, primitive, even 87
9T24 [33:2]S(3) 324
22 · 34
1 solvable, semiabelian 502
9T25 [1/2.S(3)3]3 324
22 · 34
1 solvable, semiabelian, even 218
9T26 E(9):2S4 432
24 · 33
1 solvable, primitive 169
9T27 L(9)=PSL(2,8) 504
23 · 32 · 7
1 not solvable, primitive, simple, irreducible, even 94
9T28 [S(3)3]3=S(3)wr3 648
23 · 34
1 solvable, semiabelian 725
9T29 [1/2.S(3)3]S(3) 648
23 · 34
1 solvable, semiabelian 306
9T30 1/2[S(3)3]S(3) 648
23 · 34
1 solvable, semiabelian, even 401
9T31 [S(3)3]S(3)=S(3)wrS(3) 1296
24 · 34
1 solvable, semiabelian 896
9T32 L(9):3=P|L(2,8) 1512
23 · 33 · 7
1 not solvable, primitive, irreducible, even 296
9T33 A(9) 181440
26 · 34 · 5 · 7
1 not solvable, primitive, simple, irreducible, even 222
9T34 S(9) 362880
27 · 34 · 5 · 7
1 not solvable, primitive, irreducible 1000

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