direct product, metabelian, supersoluble, monomial, rational, 2-hyperelementary
Aliases: S3xD4, C4:1D6, C12:C22, D12:3C2, C22:2D6, D6:2C22, C6.5C23, Dic3:1C22, C3:2(C2xD4), (C2xC6):C22, (C4xS3):1C2, (C3xD4):2C2, C3:D4:1C2, (C22xS3):2C2, C2.6(C22xS3), Aut(D12), Hol(C12), SmallGroup(48,38)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for S3xD4
G = < a,b,c,d | a3=b2=c4=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Subgroups: 120 in 54 conjugacy classes, 25 normal (13 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, S3, C6, C6, C2xC4, D4, D4, C23, Dic3, C12, D6, D6, D6, C2xC6, C2xD4, C4xS3, D12, C3:D4, C3xD4, C22xS3, S3xD4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2xD4, C22xS3, S3xD4
Character table of S3xD4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 6A | 6B | 6C | 12 | |
| size | 1 | 1 | 2 | 2 | 3 | 3 | 6 | 6 | 2 | 2 | 6 | 2 | 4 | 4 | 4 | |
| ρ1 | 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 | linear of order 2 |
| ρ3 | 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 | linear of order 2 |
| ρ5 | 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 | linear of order 2 |
| ρ7 | 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 | linear of order 2 |
| ρ9 | 2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | -1 | -2 | 0 | -1 | -1 | 1 | 1 | orthogonal lifted from D6 |
| ρ10 | 2 | -2 | 0 | 0 | 2 | -2 | 0 | 0 | 2 | 0 | 0 | -2 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ11 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -1 | 2 | 0 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ12 | 2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | -1 | -2 | 0 | -1 | 1 | -1 | 1 | orthogonal lifted from D6 |
| ρ13 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | -1 | 2 | 0 | -1 | 1 | 1 | -1 | orthogonal lifted from D6 |
| ρ14 | 2 | -2 | 0 | 0 | -2 | 2 | 0 | 0 | 2 | 0 | 0 | -2 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ15 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 2 | 0 | 0 | 0 | orthogonal faithful |
►On 12 points - transitive group 12T28
Generators in S12
(1 9 8)(2 10 5)(3 11 6)(4 12 7)
(5 10)(6 11)(7 12)(8 9)
(1 2 3 4)(5 6 7 8)(9 10 11 12)
(2 4)(5 7)(10 12)
G:=sub<Sym(12)| (1,9,8)(2,10,5)(3,11,6)(4,12,7), (5,10)(6,11)(7,12)(8,9), (1,2,3,4)(5,6,7,8)(9,10,11,12), (2,4)(5,7)(10,12)>;
G:=Group( (1,9,8)(2,10,5)(3,11,6)(4,12,7), (5,10)(6,11)(7,12)(8,9), (1,2,3,4)(5,6,7,8)(9,10,11,12), (2,4)(5,7)(10,12) );
G=PermutationGroup([[(1,9,8),(2,10,5),(3,11,6),(4,12,7)], [(5,10),(6,11),(7,12),(8,9)], [(1,2,3,4),(5,6,7,8),(9,10,11,12)], [(2,4),(5,7),(10,12)]])
G:=TransitiveGroup(12,28);
►On 24 points - transitive group 24T52
Generators in S24
(1 17 16)(2 18 13)(3 19 14)(4 20 15)(5 10 23)(6 11 24)(7 12 21)(8 9 22)
(1 24)(2 21)(3 22)(4 23)(5 15)(6 16)(7 13)(8 14)(9 19)(10 20)(11 17)(12 18)
(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 4)(5 7)(10 12)(13 15)(18 20)(21 23)
G:=sub<Sym(24)| (1,17,16)(2,18,13)(3,19,14)(4,20,15)(5,10,23)(6,11,24)(7,12,21)(8,9,22), (1,24)(2,21)(3,22)(4,23)(5,15)(6,16)(7,13)(8,14)(9,19)(10,20)(11,17)(12,18), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (2,4)(5,7)(10,12)(13,15)(18,20)(21,23)>;
G:=Group( (1,17,16)(2,18,13)(3,19,14)(4,20,15)(5,10,23)(6,11,24)(7,12,21)(8,9,22), (1,24)(2,21)(3,22)(4,23)(5,15)(6,16)(7,13)(8,14)(9,19)(10,20)(11,17)(12,18), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (2,4)(5,7)(10,12)(13,15)(18,20)(21,23) );
G=PermutationGroup([[(1,17,16),(2,18,13),(3,19,14),(4,20,15),(5,10,23),(6,11,24),(7,12,21),(8,9,22)], [(1,24),(2,21),(3,22),(4,23),(5,15),(6,16),(7,13),(8,14),(9,19),(10,20),(11,17),(12,18)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24)], [(2,4),(5,7),(10,12),(13,15),(18,20),(21,23)]])
G:=TransitiveGroup(24,52);
►On 24 points - transitive group 24T53
Generators in S24
(1 20 21)(2 17 22)(3 18 23)(4 19 24)(5 10 14)(6 11 15)(7 12 16)(8 9 13)
(1 3)(2 4)(5 7)(6 8)(9 15)(10 16)(11 13)(12 14)(17 24)(18 21)(19 22)(20 23)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)
(1 8)(2 7)(3 6)(4 5)(9 20)(10 19)(11 18)(12 17)(13 21)(14 24)(15 23)(16 22)
G:=sub<Sym(24)| (1,20,21)(2,17,22)(3,18,23)(4,19,24)(5,10,14)(6,11,15)(7,12,16)(8,9,13), (1,3)(2,4)(5,7)(6,8)(9,15)(10,16)(11,13)(12,14)(17,24)(18,21)(19,22)(20,23), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,8)(2,7)(3,6)(4,5)(9,20)(10,19)(11,18)(12,17)(13,21)(14,24)(15,23)(16,22)>;
G:=Group( (1,20,21)(2,17,22)(3,18,23)(4,19,24)(5,10,14)(6,11,15)(7,12,16)(8,9,13), (1,3)(2,4)(5,7)(6,8)(9,15)(10,16)(11,13)(12,14)(17,24)(18,21)(19,22)(20,23), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,8)(2,7)(3,6)(4,5)(9,20)(10,19)(11,18)(12,17)(13,21)(14,24)(15,23)(16,22) );
G=PermutationGroup([[(1,20,21),(2,17,22),(3,18,23),(4,19,24),(5,10,14),(6,11,15),(7,12,16),(8,9,13)], [(1,3),(2,4),(5,7),(6,8),(9,15),(10,16),(11,13),(12,14),(17,24),(18,21),(19,22),(20,23)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24)], [(1,8),(2,7),(3,6),(4,5),(9,20),(10,19),(11,18),(12,17),(13,21),(14,24),(15,23),(16,22)]])
G:=TransitiveGroup(24,53);
►On 24 points - transitive group 24T54
Generators in S24
(1 20 21)(2 17 22)(3 18 23)(4 19 24)(5 10 14)(6 11 15)(7 12 16)(8 9 13)
(1 6)(2 7)(3 8)(4 5)(9 23)(10 24)(11 21)(12 22)(13 18)(14 19)(15 20)(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)
(1 8)(2 7)(3 6)(4 5)(9 20)(10 19)(11 18)(12 17)(13 21)(14 24)(15 23)(16 22)
G:=sub<Sym(24)| (1,20,21)(2,17,22)(3,18,23)(4,19,24)(5,10,14)(6,11,15)(7,12,16)(8,9,13), (1,6)(2,7)(3,8)(4,5)(9,23)(10,24)(11,21)(12,22)(13,18)(14,19)(15,20)(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), (1,8)(2,7)(3,6)(4,5)(9,20)(10,19)(11,18)(12,17)(13,21)(14,24)(15,23)(16,22)>;
G:=Group( (1,20,21)(2,17,22)(3,18,23)(4,19,24)(5,10,14)(6,11,15)(7,12,16)(8,9,13), (1,6)(2,7)(3,8)(4,5)(9,23)(10,24)(11,21)(12,22)(13,18)(14,19)(15,20)(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), (1,8)(2,7)(3,6)(4,5)(9,20)(10,19)(11,18)(12,17)(13,21)(14,24)(15,23)(16,22) );
G=PermutationGroup([[(1,20,21),(2,17,22),(3,18,23),(4,19,24),(5,10,14),(6,11,15),(7,12,16),(8,9,13)], [(1,6),(2,7),(3,8),(4,5),(9,23),(10,24),(11,21),(12,22),(13,18),(14,19),(15,20),(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)], [(1,8),(2,7),(3,6),(4,5),(9,20),(10,19),(11,18),(12,17),(13,21),(14,24),(15,23),(16,22)]])
G:=TransitiveGroup(24,54);
S3xD4 is a maximal subgroup of
D8:S3 Q8:3D6 D4:6D6 D4oD12 D6:D6 Dic3:D6 D4.4S4 C20:D6 D10:D6 C28:D6 D6:D14 C4:S5 C4.3S5 C22:S5
S3xD4 is a maximal quotient of
Dic3.D4 Dic3:4D4 D6:D4 C23.9D6 Dic3:D4 C23.11D6 C12:Q8 Dic3:5D4 D6.D4 C12:D4 D6:Q8 D8:S3 D8:3S3 Q8:3D6 D4.D6 Q8.7D6 Q16:S3 D24:C2 C23:2D6 D6:3D4 C23.14D6 C12:3D4 D6:D6 Dic3:D6 C20:D6 D10:D6 C28:D6 D6:D14
| action | f(x) | Disc(f) |
|---|---|---|
| 12T28 | x12-4x6-5x3-1 | -318·56·113 |
Matrix representation of S3xD4 ►in GL4(Z) generated by
| -1 | -1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | -1 | -1 |
| 0 | 0 | 1 | 0 |
| -1 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 |
| 0 | 0 | -1 | 0 |
| 0 | 0 | 1 | 1 |
| 0 | 0 | -1 | 0 |
| 0 | 0 | 0 | -1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | -1 | 0 |
| 0 | 0 | 0 | -1 |
G:=sub<GL(4,Integers())| [-1,1,0,0,-1,0,0,0,0,0,-1,1,0,0,-1,0],[-1,1,0,0,0,1,0,0,0,0,-1,1,0,0,0,1],[0,0,1,0,0,0,0,1,-1,0,0,0,0,-1,0,0],[1,0,0,0,0,1,0,0,0,0,-1,0,0,0,0,-1] >;
S3xD4 in GAP, Magma, Sage, TeX
S_3\times D_4
% in TeX
G:=Group("S3xD4"); // GroupNames label
G:=SmallGroup(48,38);
// by ID
G=gap.SmallGroup(48,38);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-3,97,804]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^2=c^4=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
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