metabelian, supersoluble, monomial
Aliases: D12:4D6, D4:3S32, C3:S3:3D8, C3:C8:13D6, C3:3(S3xD8), D4:S3:2S3, (C3xD4):1D6, C32:8(C2xD8), C6.55(S3xD4), D6:D6:3C2, C3:D24:8C2, (C3xD12):6C22, C3:Dic3.19D4, (C3xC12).3C23, C12.3(C22xS3), C12:S3:4C22, C12.29D6:3C2, (D4xC32):3C22, C2.15(Dic3:D6), C4.3(C2xS32), (D4xC3:S3):1C2, (C3xD4:S3):2C2, (C3xC3:C8):6C22, (C2xC3:S3).56D4, (C3xC6).118(C2xD4), (C4xC3:S3).11C22, SmallGroup(288,574)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D12:D6
G = < a,b,c,d | a12=b2=c6=d2=1, bab=dad=a-1, cac-1=a7, cbc-1=a3b, dbd=a7b, dcd=c-1 >
Subgroups: 1026 in 179 conjugacy classes, 40 normal (16 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C8, C2xC4, D4, D4, C23, C32, Dic3, C12, C12, D6, C2xC6, C2xC8, D8, C2xD4, C3xS3, C3:S3, C3:S3, C3xC6, C3xC6, C3:C8, C24, C4xS3, D12, D12, C3:D4, C3xD4, C3xD4, C22xS3, C2xD8, C3:Dic3, C3xC12, S32, S3xC6, C2xC3:S3, C2xC3:S3, C62, S3xC8, D24, D4:S3, D4:S3, C3xD8, S3xD4, C3xC3:C8, D6:S3, C3xD12, C4xC3:S3, C12:S3, C32:7D4, D4xC32, C2xS32, C22xC3:S3, S3xD8, C12.29D6, C3:D24, C3xD4:S3, D6:D6, D4xC3:S3, D12:D6
Quotients: C1, C2, C22, S3, D4, C23, D6, D8, C2xD4, C22xS3, C2xD8, S32, S3xD4, C2xS32, S3xD8, Dic3:D6, D12:D6
►On 24 points - transitive group 24T668
(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 21)(2 20)(3 19)(4 18)(5 17)(6 16)(7 15)(8 14)(9 13)(10 24)(11 23)(12 22)
(1 9 5)(2 4 6 8 10 12)(3 11 7)(13 14 21 22 17 18)(15 16 23 24 19 20)
(1 5)(2 4)(6 12)(7 11)(8 10)(13 18)(14 17)(15 16)(19 24)(20 23)(21 22)
G:=sub<Sym(24)| (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24), (1,21)(2,20)(3,19)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)(10,24)(11,23)(12,22), (1,9,5)(2,4,6,8,10,12)(3,11,7)(13,14,21,22,17,18)(15,16,23,24,19,20), (1,5)(2,4)(6,12)(7,11)(8,10)(13,18)(14,17)(15,16)(19,24)(20,23)(21,22)>;
G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24), (1,21)(2,20)(3,19)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)(10,24)(11,23)(12,22), (1,9,5)(2,4,6,8,10,12)(3,11,7)(13,14,21,22,17,18)(15,16,23,24,19,20), (1,5)(2,4)(6,12)(7,11)(8,10)(13,18)(14,17)(15,16)(19,24)(20,23)(21,22) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12),(13,14,15,16,17,18,19,20,21,22,23,24)], [(1,21),(2,20),(3,19),(4,18),(5,17),(6,16),(7,15),(8,14),(9,13),(10,24),(11,23),(12,22)], [(1,9,5),(2,4,6,8,10,12),(3,11,7),(13,14,21,22,17,18),(15,16,23,24,19,20)], [(1,5),(2,4),(6,12),(7,11),(8,10),(13,18),(14,17),(15,16),(19,24),(20,23),(21,22)]])
G:=TransitiveGroup(24,668);
33 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 3C | 4A | 4B | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 6I | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 24A | 24B | 24C | 24D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 24 | 24 | 24 | 24 |
| size | 1 | 1 | 4 | 9 | 9 | 12 | 12 | 36 | 2 | 2 | 4 | 2 | 18 | 2 | 2 | 4 | 8 | 8 | 8 | 8 | 24 | 24 | 6 | 6 | 6 | 6 | 4 | 4 | 8 | 12 | 12 | 12 | 12 |
33 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 8 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D6 | D6 | D6 | D8 | S32 | S3xD4 | C2xS32 | S3xD8 | Dic3:D6 | D12:D6 |
| kernel | D12:D6 | C12.29D6 | C3:D24 | C3xD4:S3 | D6:D6 | D4xC3:S3 | D4:S3 | C3:Dic3 | C2xC3:S3 | C3:C8 | D12 | C3xD4 | C3:S3 | D4 | C6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 2 | 4 | 1 | 2 | 1 | 4 | 2 | 1 |
Matrix representation of D12:D6 ►in GL6(F73)
| 0 | 1 | 0 | 0 | 0 | 0 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 1 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 16 | 16 | 0 | 0 | 0 | 0 |
| 16 | 57 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 72 | 72 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(6,GF(73))| [0,72,0,0,0,0,1,0,0,0,0,0,0,0,72,72,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[16,16,0,0,0,0,16,57,0,0,0,0,0,0,1,1,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,0,1,72],[1,0,0,0,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,0,72,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;
D12:D6 in GAP, Magma, Sage, TeX
D_{12}\rtimes D_6 % in TeX
G:=Group("D12:D6"); // GroupNames label
G:=SmallGroup(288,574);
// by ID
G=gap.SmallGroup(288,574);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,254,135,675,346,185,80,1356,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^12=b^2=c^6=d^2=1,b*a*b=d*a*d=a^-1,c*a*c^-1=a^7,c*b*c^-1=a^3*b,d*b*d=a^7*b,d*c*d=c^-1>;
// generators/relations