direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: S3xD8, C8:4D6, D4:1D6, D24:4C2, C24:2C22, D6.12D4, D12:1C22, C12.1C23, Dic3.3D4, C3:2(C2xD8), (S3xC8):1C2, D4:S3:1C2, (C3xD8):2C2, (S3xD4):1C2, C3:C8:5C22, C2.15(S3xD4), C6.27(C2xD4), (C3xD4):1C22, C4.1(C22xS3), (C4xS3).7C22, SmallGroup(96,117)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for S3xD8
G = < a,b,c,d | a3=b2=c8=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Subgroups: 242 in 76 conjugacy classes, 29 normal (17 characteristic)
C1, C2, C2, C3, C4, C4, C22, S3, S3, C6, C6, C8, C8, C2xC4, D4, D4, C23, Dic3, C12, D6, D6, C2xC6, C2xC8, D8, D8, C2xD4, C3:C8, C24, C4xS3, D12, C3:D4, C3xD4, C22xS3, C2xD8, S3xC8, D24, D4:S3, C3xD8, S3xD4, S3xD8
Quotients: C1, C2, C22, S3, D4, C23, D6, D8, C2xD4, C22xS3, C2xD8, S3xD4, S3xD8
Character table of S3xD8
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 12 | 24A | 24B | |
| size | 1 | 1 | 3 | 3 | 4 | 4 | 12 | 12 | 2 | 2 | 6 | 2 | 8 | 8 | 2 | 2 | 6 | 6 | 4 | 4 | 4 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | -1 | -1 | 1 | -1 | -1 | linear of order 2 |
| ρ3 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | linear of order 2 |
| ρ4 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | linear of order 2 |
| ρ5 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | 1 | -1 | -1 | linear of order 2 |
| ρ6 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | linear of order 2 |
| ρ7 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | linear of order 2 |
| ρ8 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ9 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 2 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | orthogonal lifted from D4 |
| ρ11 | 2 | 2 | 0 | 0 | -2 | 2 | 0 | 0 | -1 | 2 | 0 | -1 | 1 | -1 | -2 | -2 | 0 | 0 | -1 | 1 | 1 | orthogonal lifted from D6 |
| ρ12 | 2 | 2 | 0 | 0 | -2 | -2 | 0 | 0 | -1 | 2 | 0 | -1 | 1 | 1 | 2 | 2 | 0 | 0 | -1 | -1 | -1 | orthogonal lifted from D6 |
| ρ13 | 2 | 2 | 0 | 0 | 2 | -2 | 0 | 0 | -1 | 2 | 0 | -1 | -1 | 1 | -2 | -2 | 0 | 0 | -1 | 1 | 1 | orthogonal lifted from D6 |
| ρ14 | 2 | 2 | 0 | 0 | 2 | 2 | 0 | 0 | -1 | 2 | 0 | -1 | -1 | -1 | 2 | 2 | 0 | 0 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ15 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | -2 | 0 | 0 | √2 | -√2 | √2 | -√2 | 0 | √2 | -√2 | orthogonal lifted from D8 |
| ρ16 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | -2 | 0 | 0 | -√2 | √2 | √2 | -√2 | 0 | -√2 | √2 | orthogonal lifted from D8 |
| ρ17 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | -2 | 0 | 0 | √2 | -√2 | -√2 | √2 | 0 | √2 | -√2 | orthogonal lifted from D8 |
| ρ18 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | -2 | 0 | 0 | -√2 | √2 | -√2 | √2 | 0 | -√2 | √2 | orthogonal lifted from D8 |
| ρ19 | 4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -4 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | orthogonal lifted from S3xD4 |
| ρ20 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 2 | 0 | 0 | -2√2 | 2√2 | 0 | 0 | 0 | √2 | -√2 | orthogonal faithful |
| ρ21 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 2 | 0 | 0 | 2√2 | -2√2 | 0 | 0 | 0 | -√2 | √2 | orthogonal faithful |
►On 24 points - transitive group 24T139
(1 23 10)(2 24 11)(3 17 12)(4 18 13)(5 19 14)(6 20 15)(7 21 16)(8 22 9)
(1 5)(2 6)(3 7)(4 8)(9 18)(10 19)(11 20)(12 21)(13 22)(14 23)(15 24)(16 17)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)
(2 8)(3 7)(4 6)(9 11)(12 16)(13 15)(17 21)(18 20)(22 24)
G:=sub<Sym(24)| (1,23,10)(2,24,11)(3,17,12)(4,18,13)(5,19,14)(6,20,15)(7,21,16)(8,22,9), (1,5)(2,6)(3,7)(4,8)(9,18)(10,19)(11,20)(12,21)(13,22)(14,23)(15,24)(16,17), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24), (2,8)(3,7)(4,6)(9,11)(12,16)(13,15)(17,21)(18,20)(22,24)>;
G:=Group( (1,23,10)(2,24,11)(3,17,12)(4,18,13)(5,19,14)(6,20,15)(7,21,16)(8,22,9), (1,5)(2,6)(3,7)(4,8)(9,18)(10,19)(11,20)(12,21)(13,22)(14,23)(15,24)(16,17), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24), (2,8)(3,7)(4,6)(9,11)(12,16)(13,15)(17,21)(18,20)(22,24) );
G=PermutationGroup([[(1,23,10),(2,24,11),(3,17,12),(4,18,13),(5,19,14),(6,20,15),(7,21,16),(8,22,9)], [(1,5),(2,6),(3,7),(4,8),(9,18),(10,19),(11,20),(12,21),(13,22),(14,23),(15,24),(16,17)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24)], [(2,8),(3,7),(4,6),(9,11),(12,16),(13,15),(17,21),(18,20),(22,24)]])
G:=TransitiveGroup(24,139);
S3xD8 is a maximal subgroup of
D8:D6 D48:C2 D8:13D6 D8:15D6 D8:5D6 C24:4D6 D12:D6 C40:5D6 D15:D8
S3xD8 is a maximal quotient of
Dic3:4D8 Dic3.D8 Dic3.SD16 D4:D12 D6.D8 D6:D8 D12:3D4 Dic3:5D8 C24:2Q8 D6.5D8 D6:2D8 D12:2Q8 D8:D6 D16:3S3 D48:C2 SD32:S3 D6.2D8 Q32:S3 D48:5C2 Dic3:D8 C24:5D4 D12:D4 D6:3D8 C24:4D6 D12:D6 C40:5D6 D15:D8
Matrix representation of S3xD8 ►in GL4(F7) generated by
| 3 | 0 | 4 | 6 |
| 1 | 2 | 6 | 5 |
| 4 | 0 | 2 | 4 |
| 1 | 0 | 3 | 5 |
| 5 | 3 | 4 | 3 |
| 0 | 6 | 4 | 4 |
| 4 | 2 | 0 | 5 |
| 3 | 5 | 1 | 3 |
| 6 | 3 | 0 | 2 |
| 1 | 6 | 1 | 1 |
| 2 | 2 | 2 | 3 |
| 3 | 4 | 2 | 6 |
| 2 | 3 | 1 | 4 |
| 3 | 6 | 0 | 3 |
| 6 | 2 | 2 | 2 |
| 6 | 4 | 6 | 4 |
G:=sub<GL(4,GF(7))| [3,1,4,1,0,2,0,0,4,6,2,3,6,5,4,5],[5,0,4,3,3,6,2,5,4,4,0,1,3,4,5,3],[6,1,2,3,3,6,2,4,0,1,2,2,2,1,3,6],[2,3,6,6,3,6,2,4,1,0,2,6,4,3,2,4] >;
S3xD8 in GAP, Magma, Sage, TeX
S_3\times D_8
% in TeX
G:=Group("S3xD8"); // GroupNames label
G:=SmallGroup(96,117);
// by ID
G=gap.SmallGroup(96,117);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-3,116,297,159,69,2309]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^2=c^8=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations
Export