Transitive Subgroups of S8

TName(s)|G|Parity|CS8(G)|SubfieldsOther RepresentationsResolvents
1C(8)=88 = 23-182T1, 4T1  2: 2T1
4: 4T1
24[x]28 = 23182T1, 2T1, 2T1, 4T1, 4T1, 4T2  2: 2T1, 2T1, 2T1
4: 4T1, 4T1, 4T2
3E(8)=2[x]2[x]28 = 23182T1, 2T1, 2T1, 2T1, 2T1, 2T1, 2T1, 4T2, 4T2, 4T2, 4T2, 4T2, 4T2, 4T2  2: 2T1, 2T1, 2T1, 2T1, 2T1, 2T1, 2T1
4: 4T2, 4T2, 4T2, 4T2, 4T2, 4T2, 4T2
4D_8(8)=[4]28 = 23182T1, 2T1, 2T1, 4T2, 4T3, 4T3 4T3, 4T32: 2T1, 2T1, 2T1
4: 4T2
5Q_8(8)8 = 23182T1, 2T1, 2T1, 4T2  2: 2T1, 2T1, 2T1
4: 4T2
6D(8)16 = 24-122T1, 4T3 8T6, 16T132: 2T1, 2T1, 2T1
4: 4T2
8: 4T3
71/2[2^3]416 = 24-142T1, 4T1 16T62: 2T1, 2T1, 2T1
4: 4T1, 4T1, 4T2
8: 8T2
82D_8(8)=[D(4)]216 = 24-122T1, 4T3 16T122: 2T1, 2T1, 2T1
4: 4T2
8: 4T3
9E(8):2=D(4)[x]216 = 24142T1, 2T1, 2T1, 4T2, 4T3, 4T3 8T9, 8T9, 8T9, 16T92: 2T1, 2T1, 2T1, 2T1, 2T1, 2T1, 2T1
4: 4T2, 4T2, 4T2, 4T2, 4T2, 4T2, 4T2
8: 4T3, 4T3, 8T3
10[2^2]416 = 24142T1, 4T1, 4T3, 4T3 8T10, 16T102: 2T1, 2T1, 2T1
4: 4T1, 4T1, 4T2
8: 4T3, 4T3, 8T2
111/2[2^3]E(4)=Q_8:216 = 24142T1, 2T1, 2T1, 4T2 8T11, 8T11, 16T112: 2T1, 2T1, 2T1, 2T1, 2T1, 2T1, 2T1
4: 4T2, 4T2, 4T2, 4T2, 4T2, 4T2, 4T2
8: 8T3
122A_4(8)=[2]A(4)=SL(2,3)24 = 23 · 3124T4  3: 3T1
12: 4T4
13E(8):3=A(4)[x]224 = 23 · 3122T1, 4T4 6T6, 12T6, 12T72: 2T1
3: 3T1
6: 6T1
12: 4T4
14S(4)[1/2]2=1/2(S_4[x]2)24 = 23 · 3122T1, 4T5 4T5, 6T7, 6T8, 12T8, 12T92: 2T1
6: 3T2
15[1/4.cD(4)^2]232 = 25-122T1, 4T3 8T15, 16T35, 16T38, 16T38, 16T452: 2T1, 2T1, 2T1, 2T1, 2T1, 2T1, 2T1
4: 4T2, 4T2, 4T2, 4T2, 4T2, 4T2, 4T2
8: 4T3, 4T3, 8T3
16: 8T9
161/2[2^4]432 = 25-122T1, 4T1 8T16, 16T36, 16T41, 16T412: 2T1, 2T1, 2T1
4: 4T1, 4T1, 4T2
8: 4T3, 4T3, 8T2
16: 8T10
17[4^2]232 = 25-142T1, 4T3 8T17, 16T28, 16T422: 2T1, 2T1, 2T1
4: 4T1, 4T1, 4T2
8: 4T3, 4T3, 8T2
16: 8T10
18E(8):E_4=[2^2]D(4)32 = 25142T1, 4T3, 4T3, 4T3 8T18, 8T18, 8T18, 8T18, 8T18, 8T18, 8T18, 16T39, 16T39, 16T39, 16T39, 16T39, 16T39, 16T462: 2T1, 2T1, 2T1, 2T1, 2T1, 2T1, 2T1
4: 4T2, 4T2, 4T2, 4T2, 4T2, 4T2, 4T2
8: 4T3, 4T3, 4T3, 4T3, 4T3, 4T3, 8T3
16: 8T9, 8T9, 8T9
19E(8):4=[1/4.eD(4)^2]232 = 25122T1, 4T3 8T19, 8T20, 8T21, 16T33, 16T33, 16T52, 16T532: 2T1, 2T1, 2T1
4: 4T1, 4T1, 4T2
8: 4T3, 4T3, 8T2
16: 8T10
20[2^3]432 = 25122T1, 4T1 8T19, 8T19, 8T21, 16T33, 16T33, 16T52, 16T532: 2T1, 2T1, 2T1
4: 4T1, 4T1, 4T2
8: 4T3, 4T3, 8T2
16: 8T10
211/2[2^4]E(4)=[1/4.dD(4)^2]232 = 25-122T1, 2T1, 2T1, 4T2 8T19, 8T19, 8T20, 16T33, 16T33, 16T52, 16T532: 2T1, 2T1, 2T1
4: 4T1, 4T1, 4T2
8: 4T3, 4T3, 8T2
16: 8T10
22E(8):D_4=[2^3]2^232 = 25122T1, 2T1, 2T1, 4T2 8T22, 8T22, 8T22, 8T22, 8T22, 16T23, 16T23, 16T23, 16T23, 16T23, 16T23, 16T23, 16T23, 16T232: 2T1, 2T1, 2T1, 2T1, 2T1, 2T1, 2T1, 2T1, 2T1, 2T1, 2T1, 2T1, 2T1, 2T1, 2T1
4: 4T2, 4T2, 4T2, 4T2, 4T2, 4T2, 4T2, 4T2, 4T2, 4T2, 4T2, 4T2, 4T2, 4T2, 4T2, 4T2, 4T2, 4T2, 4T2, 4T2, 4T2, 4T2, 4T2, 4T2, 4T2, 4T2, 4T2, 4T2, 4T2, 4T2, 4T2, 4T2, 4T2, 4T2, 4T2
8: 8T3, 8T3, 8T3, 8T3, 8T3, 8T3, 8T3, 8T3, 8T3, 8T3, 8T3, 8T3, 8T3, 8T3, 8T3
16: 16T3
232S_4(8)=GL(2,3)48 = 24 · 3-124T5 8T23, 16T662: 2T1
6: 3T2
24: 4T5
24E(8):D_6=S(4)[x]248 = 24 · 3122T1, 4T5 6T11, 6T11, 8T24, 12T21, 12T22, 12T23, 12T23, 12T24, 12T24, 16T612: 2T1, 2T1, 2T1
4: 4T2
6: 3T2
12: 6T3
24: 4T5
25E(8):7=F_56(8)56 = 23 · 711  14T67: 7T1
261/2[2^4]eD(4)64 = 26-122T1, 4T3 8T26, 8T26, 8T26, 16T135, 16T135, 16T141, 16T141, 16T142, 16T142, 16T152, 16T1522: 2T1, 2T1, 2T1, 2T1, 2T1, 2T1, 2T1
4: 4T2, 4T2, 4T2, 4T2, 4T2, 4T2, 4T2
8: 4T3, 4T3, 4T3, 4T3, 4T3, 4T3, 8T3
16: 8T9, 8T9, 8T9
32: 8T18
27[2^4]464 = 26-122T1, 4T1 8T27, 8T28, 8T28, 16T130, 16T157, 16T157, 16T158, 16T158, 16T159, 16T159, 16T166, 16T170, 16T171, 16T1722: 2T1, 2T1, 2T1
4: 4T1, 4T1, 4T2
8: 4T3, 4T3, 8T2
16: 8T10
32: 8T19
281/2[2^4]dD(4)64 = 26-122T1, 4T3 8T27, 8T27, 8T28, 16T130, 16T157, 16T157, 16T158, 16T158, 16T159, 16T159, 16T166, 16T170, 16T171, 16T1722: 2T1, 2T1, 2T1
4: 4T1, 4T1, 4T2
8: 4T3, 4T3, 8T2
16: 8T10
32: 8T19
29E(8):D_8=[2^3]D(4)64 = 26122T1, 4T3 8T29, 8T29, 8T29, 8T29, 8T29, 8T31, 8T31, 16T127, 16T128, 16T128, 16T128, 16T129, 16T129, 16T129, 16T147, 16T149, 16T149, 16T149, 16T149, 16T149, 16T149, 16T150, 16T150, 16T1502: 2T1, 2T1, 2T1, 2T1, 2T1, 2T1, 2T1
4: 4T2, 4T2, 4T2, 4T2, 4T2, 4T2, 4T2
8: 4T3, 4T3, 4T3, 4T3, 4T3, 4T3, 8T3
16: 8T9, 8T9, 8T9
32: 8T18
301/2[2^4]cD(4)64 = 26-122T1, 4T3 8T30, 8T30, 8T30, 16T143, 16T143, 16T167, 16T167, 16T168, 16T168, 16T169, 16T1692: 2T1, 2T1, 2T1
4: 4T1, 4T1, 4T2
8: 4T3, 4T3, 8T2
16: 8T10
32: 8T19
31[2^4]E(4)64 = 26-122T1, 2T1, 2T1, 4T2 8T29, 8T29, 8T29, 8T29, 8T29, 8T29, 8T31, 16T127, 16T128, 16T128, 16T128, 16T129, 16T129, 16T129, 16T147, 16T149, 16T149, 16T149, 16T149, 16T149, 16T149, 16T150, 16T150, 16T1502: 2T1, 2T1, 2T1, 2T1, 2T1, 2T1, 2T1
4: 4T2, 4T2, 4T2, 4T2, 4T2, 4T2, 4T2
8: 4T3, 4T3, 4T3, 4T3, 4T3, 4T3, 8T3
16: 8T9, 8T9, 8T9
32: 8T18
32[2^3]A(4)96 = 25 · 3124T4 8T32, 8T323: 3T1
12: 4T4, 4T4, 4T4, 4T4, 4T4
48: 12T32
33E(8):A_4=[1/3.A(4)^2]2=E(4):696 = 25 · 3112T1 8T33, 12T58, 12T58, 12T59, 12T59, 16T1832: 2T1
3: 3T1
6: 6T1
12: 4T4
24: 6T6
341/2[E(4)^2:S_3]2=E(4)^2:D_696 = 25 · 3112T1 12T66, 12T66, 12T66, 12T67, 12T68, 12T68, 12T68, 12T69, 16T1942: 2T1
6: 3T2
24: 4T5, 4T5, 4T5
35[2^4]D(4)128 = 27-122T1, 4T3 8T35, 8T35, 8T35, 8T35, 8T35, 8T35, 8T35, 16T376, 16T376, 16T376, 16T376, 16T388, 16T388, 16T388, 16T388, 16T390, 16T390, 16T390, 16T390, 16T391, 16T391, 16T391, 16T391, 16T393, 16T393, 16T393, 16T393, 16T395, 16T395, 16T395, 16T395, 16T396, 16T396, 16T396, 16T396, 16T401, 16T401, 16T401, 16T4012: 2T1, 2T1, 2T1, 2T1, 2T1, 2T1, 2T1
4: 4T2, 4T2, 4T2, 4T2, 4T2, 4T2, 4T2
8: 4T3, 4T3, 4T3, 4T3, 4T3, 4T3, 8T3
16: 8T9, 8T9, 8T9
32: 8T18
64: 8T29
36E(8):F_21168 = 23 · 3 · 711  14T113: 3T1
21: 7T3
37L(8)=PSL(2,7)168 = 23 · 3 · 711  7T5, 7T5, 14T10, 14T10, 21T14 
38[2^4]A(4)192 = 26 · 3-124T4 8T38, 16T425, 16T4272: 2T1
3: 3T1
6: 6T1
12: 4T4
24: 6T6
96: 8T33
39[2^3]S(4)192 = 26 · 3124T5 8T39, 8T39, 8T39, 8T39, 8T39, 16T442, 16T442, 16T4422: 2T1
6: 3T2
24: 4T5, 4T5, 4T5
96: 8T34
401/2[2^4]S(4)192 = 26 · 3-124T5 8T40, 16T444, 16T4452: 2T1
6: 3T2
24: 4T5, 4T5, 4T5
96: 8T34
41E(8):S_4=[E(4)^2:S_3]2=E(4)^2:D_12192 = 26 · 3112T1 8T41, 12T108, 12T108, 12T109, 12T109, 12T110, 12T110, 12T111, 12T111, 16T435, 16T435, 16T4362: 2T1, 2T1, 2T1
4: 4T2
6: 3T2
12: 6T3
24: 4T5
48: 6T11
42[A(4)^2]2288 = 25 · 32112T1 12T126, 12T128, 12T129, 16T708, 18T112, 18T1132: 2T1
3: 3T1
6: 3T2, 6T1
18: 6T5
43L(8):2=PGL(2,7)336 = 24 · 3 · 7-11  14T16, 16T713, 21T202: 2T1
44[2^4]S(4)384 = 27 · 3-124T5 8T44, 8T44, 8T44, 16T736, 16T736, 16T743, 16T743, 16T748, 16T748, 16T752, 16T7522: 2T1, 2T1, 2T1
4: 4T2
6: 3T2
12: 6T3
24: 4T5
48: 6T11
192: 8T41
45[1/2.S(4)^2]2576 = 26 · 32112T1 12T161, 12T163, 12T165, 12T165, 16T1032, 16T1034, 18T179, 18T180, 18T185, 18T1852: 2T1, 2T1, 2T1
4: 4T2
6: 3T2, 3T2
12: 6T3, 6T3
36: 6T9
461/2[S(4)^2]2576 = 26 · 32-112T1 12T160, 12T162, 16T1030, 16T1031, 18T182, 18T1842: 2T1
4: 4T1
36: 6T10
47[S(4)^2]21152 = 27 · 32-112T1 12T200, 12T201, 12T202, 12T203, 16T1292, 16T1294, 16T1295, 16T1296, 18T272, 18T273, 18T274, 18T2752: 2T1, 2T1, 2T1
4: 4T2
8: 4T3
72: 6T13
48E(8):L_7=AL(8)1344 = 26 · 3 · 711  8T48, 14T34, 14T34168: 7T5
49A820160 = 26 · 32 · 5 · 711  15T72, 15T72 
50S840320 = 27 · 32 · 5 · 7-11  16T18382: 2T1