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Transitive Groups of degree 8

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Transitive Groups of degree 8
G Name |G| |G| fact. |Z(G)| Properties of G # fields
8T1 C(8)=8 8
23
8 cyclic, semiabelian 352
8T2 4[x]2 8
23
8 abelian, semiabelian, even 1487
8T3 E(8)=2[x]2[x]2 8
23
8 abelian, semiabelian, even 1056
8T4 D8(8)=[4]2 8
23
2 nilpotent, semiabelian, even 1403
8T5 Q8(8) 8
23
2 nilpotent, semiabelian, even 686
8T6 D(8) 16
24
2 nilpotent, semiabelian 766
8T7 1/2[23]4 16
24
4 nilpotent, semiabelian 1109
8T8 2D8(8)=[D(4)]2 16
24
2 nilpotent, semiabelian 584
8T9 E(8):2=D(4)[x]2 16
24
4 nilpotent, semiabelian, even 3844
8T10 [22]4 16
24
4 nilpotent, semiabelian, even 2719
8T11 1/2[23]E(4)=Q8:2 16
24
4 nilpotent, semiabelian, even 2134
8T12 2A4(8)=[2]A(4)=SL(2,3) 24
23 · 3
2 solvable, irreducible, even 390
8T13 E(8):3=A(4)[x]2 24
23 · 3
2 solvable, semiabelian, even 1604
8T14 S(4)[1/2]2=1/2(S4[x]2) 24
23 · 3
1 solvable, semiabelian, even 861
8T15 [1/4.cD(4)2]2 32
25
2 nilpotent, semiabelian 2249
8T16 1/2[24]4 32
25
2 nilpotent, semiabelian 2077
8T17 [42]2 32
25
4 nilpotent, semiabelian 1939
8T18 E(8):E4=[22]D(4) 32
25
4 nilpotent, semiabelian, even 2990
8T19 E(8):4=[1/4.eD(4)2]2 32
25
2 nilpotent, semiabelian, even 2787
8T20 [23]4 32
25
2 nilpotent, semiabelian, even 2405
8T21 1/2[24]E(4)=[1/4.dD(4)2]2 32
25
2 nilpotent, semiabelian 2323
8T22 E(8):D4=[23]22 32
25
2 nilpotent, semiabelian, even 2509
8T23 2S4(8)=GL(2,3) 48
24 · 3
2 solvable, irreducible 801
8T24 E(8):D6=S(4)[x]2 48
24 · 3
2 solvable, semiabelian, even 3207
8T25 E(8):7=F56(8) 56
23 · 7
1 solvable, primitive, semiabelian, even 175
8T26 1/2[24]eD(4) 64
26
2 nilpotent, semiabelian 3558
8T27 [24]4 64
26
2 nilpotent, semiabelian 3787
8T28 1/2[24]dD(4) 64
26
2 nilpotent, semiabelian 3571
8T29 E(8):D8=[23]D(4) 64
26
2 nilpotent, semiabelian, even 4236
8T30 1/2[24]cD(4) 64
26
2 nilpotent, semiabelian 2962
8T31 [24]E(4) 64
26
2 nilpotent, semiabelian 3362
8T32 [23]A(4) 96
25 · 3
2 solvable, semiabelian, even 1773
8T33 E(8):A4=[1/3.A(4)2]2=E(16):6 96
25 · 3
1 solvable, semiabelian, even 713
8T34 1/2[E(4)2:S3]2=E(4)2:D6 96
25 · 3
1 solvable, semiabelian, even 629
8T35 [24]D(4) 128
27
2 nilpotent, semiabelian 5807
8T36 E(8):F21 168
23 · 3 · 7
1 solvable, primitive, semiabelian, even 165
8T37 L(8)=PSL(2,7) 168
23 · 3 · 7
1 not solvable, primitive, simple, irreducible, even 234
8T38 [24]A(4) 192
26 · 3
2 solvable, semiabelian 3049
8T39 [23]S(4) 192
26 · 3
2 solvable, semiabelian, even 3157
8T40 1/2[24]S(4) 192
26 · 3
2 solvable 2271
8T41 E(8):S4=[E(4)2:S3]2=E(4)2:D12 192
26 · 3
1 solvable, semiabelian, even 1837
8T42 [A(4)2]2 288
25 · 32
1 solvable, semiabelian, even 559
8T43 L(8):2=PGL(2,7) 336
24 · 3 · 7
1 not solvable, primitive, irreducible 290
8T44 [24]S(4) 384
27 · 3
2 solvable, semiabelian 4964
8T45 [1/2.S(4)2]2 576
26 · 32
1 solvable, semiabelian, even 756
8T46 1/2[S(4)2]2 576
26 · 32
1 solvable, semiabelian 429
8T47 [S(4)2]2 1152
27 · 32
1 solvable, semiabelian 3155
8T48 E(8):L7=AL(8) 1344
26 · 3 · 7
1 not solvable, primitive, even 941
8T49 A(8) 20160
26 · 32 · 5 · 7
1 not solvable, primitive, simple, irreducible, even 545
8T50 S(8) 40320
27 · 32 · 5 · 7
1 not solvable, primitive, irreducible 815

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