metabelian, supersoluble, monomial
Aliases: D12:24D6, Dic6:23D6, C32:12+ 1+4, C62.139C23, (C4xS3):1D6, (C2xC12):5D6, C4oD12:8S3, (S3xD12):7C2, C3:D4:12D6, (C6xD12):17C2, (C2xD12):14S3, C3:1(D4oD12), (C6xC12):7C22, (C22xS3):5D6, D12:5S3:8C2, D12:S3:8C2, D6:D6:10C2, C3:1(D4:6D6), (S3xC12):2C22, (S3xC6).6C23, C6.14(S3xC23), (C3xC6).14C24, D6.7(C22xS3), C12.59D6:9C2, D6:S3:3C22, C3:D12:1C22, (C3xD12):31C22, (S3xDic3):1C22, C32:7D4:7C22, C12:S3:23C22, C12.109(C22xS3), (C3xC12).119C23, (C3xDic6):30C22, C3:Dic3.18C23, (C3xDic3).9C23, Dic3.6(C22xS3), C32:4Q8:22C22, (C2xC4):3S32, C4.80(C2xS32), (C2xS32):2C22, (S3xC3:D4):1C2, (S3xC2xC6):8C22, C22.8(C2xS32), (C4xC3:S3):1C22, C2.16(C22xS32), (C3xC4oD12):13C2, (C3xC3:D4):8C22, (C2xC3:S3).20C23, (C2xC6).155(C22xS3), SmallGroup(288,955)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D12:24D6
G = < a,b,c,d | a12=b2=c6=d2=1, bab=cac-1=a-1, dad=a5, cbc-1=a4b, dbd=a10b, dcd=c-1 >
Subgroups: 1434 in 352 conjugacy classes, 108 normal (36 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C22, S3, C6, C6, C2xC4, C2xC4, D4, Q8, C23, C32, Dic3, Dic3, C12, C12, D6, D6, C2xC6, C2xC6, C2xD4, C4oD4, C3xS3, C3:S3, C3xC6, C3xC6, Dic6, Dic6, C4xS3, C4xS3, D12, D12, D12, C2xDic3, C3:D4, C3:D4, C2xC12, C2xC12, C3xD4, C3xQ8, C22xS3, C22xS3, C22xC6, 2+ 1+4, C3xDic3, C3:Dic3, C3xC12, S32, S3xC6, S3xC6, C2xC3:S3, C62, C2xD12, C2xD12, C4oD12, C4oD12, S3xD4, D4:2S3, Q8:3S3, C2xC3:D4, C6xD4, C3xC4oD4, S3xDic3, D6:S3, C3:D12, C3xDic6, S3xC12, C3xD12, C3xD12, C3xC3:D4, C32:4Q8, C4xC3:S3, C12:S3, C32:7D4, C6xC12, C2xS32, S3xC2xC6, D4:6D6, D4oD12, D12:5S3, D12:S3, S3xD12, D6:D6, S3xC3:D4, C6xD12, C3xC4oD12, C12.59D6, D12:24D6
Quotients: C1, C2, C22, S3, C23, D6, C24, C22xS3, 2+ 1+4, S32, S3xC23, C2xS32, D4:6D6, D4oD12, C22xS32, D12:24D6
►On 48 points
(1 2 3 4 5 6 7 8 9 10 11 12)(13 14 15 16 17 18 19 20 21 22 23 24)(25 26 27 28 29 30 31 32 33 34 35 36)(37 38 39 40 41 42 43 44 45 46 47 48)
(1 13)(2 24)(3 23)(4 22)(5 21)(6 20)(7 19)(8 18)(9 17)(10 16)(11 15)(12 14)(25 45)(26 44)(27 43)(28 42)(29 41)(30 40)(31 39)(32 38)(33 37)(34 48)(35 47)(36 46)
(1 38 5 46 9 42)(2 37 6 45 10 41)(3 48 7 44 11 40)(4 47 8 43 12 39)(13 28 21 32 17 36)(14 27 22 31 18 35)(15 26 23 30 19 34)(16 25 24 29 20 33)
(1 35)(2 28)(3 33)(4 26)(5 31)(6 36)(7 29)(8 34)(9 27)(10 32)(11 25)(12 30)(13 37)(14 42)(15 47)(16 40)(17 45)(18 38)(19 43)(20 48)(21 41)(22 46)(23 39)(24 44)
G:=sub<Sym(48)| (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36)(37,38,39,40,41,42,43,44,45,46,47,48), (1,13)(2,24)(3,23)(4,22)(5,21)(6,20)(7,19)(8,18)(9,17)(10,16)(11,15)(12,14)(25,45)(26,44)(27,43)(28,42)(29,41)(30,40)(31,39)(32,38)(33,37)(34,48)(35,47)(36,46), (1,38,5,46,9,42)(2,37,6,45,10,41)(3,48,7,44,11,40)(4,47,8,43,12,39)(13,28,21,32,17,36)(14,27,22,31,18,35)(15,26,23,30,19,34)(16,25,24,29,20,33), (1,35)(2,28)(3,33)(4,26)(5,31)(6,36)(7,29)(8,34)(9,27)(10,32)(11,25)(12,30)(13,37)(14,42)(15,47)(16,40)(17,45)(18,38)(19,43)(20,48)(21,41)(22,46)(23,39)(24,44)>;
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)(25,26,27,28,29,30,31,32,33,34,35,36)(37,38,39,40,41,42,43,44,45,46,47,48), (1,13)(2,24)(3,23)(4,22)(5,21)(6,20)(7,19)(8,18)(9,17)(10,16)(11,15)(12,14)(25,45)(26,44)(27,43)(28,42)(29,41)(30,40)(31,39)(32,38)(33,37)(34,48)(35,47)(36,46), (1,38,5,46,9,42)(2,37,6,45,10,41)(3,48,7,44,11,40)(4,47,8,43,12,39)(13,28,21,32,17,36)(14,27,22,31,18,35)(15,26,23,30,19,34)(16,25,24,29,20,33), (1,35)(2,28)(3,33)(4,26)(5,31)(6,36)(7,29)(8,34)(9,27)(10,32)(11,25)(12,30)(13,37)(14,42)(15,47)(16,40)(17,45)(18,38)(19,43)(20,48)(21,41)(22,46)(23,39)(24,44) );
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),(25,26,27,28,29,30,31,32,33,34,35,36),(37,38,39,40,41,42,43,44,45,46,47,48)], [(1,13),(2,24),(3,23),(4,22),(5,21),(6,20),(7,19),(8,18),(9,17),(10,16),(11,15),(12,14),(25,45),(26,44),(27,43),(28,42),(29,41),(30,40),(31,39),(32,38),(33,37),(34,48),(35,47),(36,46)], [(1,38,5,46,9,42),(2,37,6,45,10,41),(3,48,7,44,11,40),(4,47,8,43,12,39),(13,28,21,32,17,36),(14,27,22,31,18,35),(15,26,23,30,19,34),(16,25,24,29,20,33)], [(1,35),(2,28),(3,33),(4,26),(5,31),(6,36),(7,29),(8,34),(9,27),(10,32),(11,25),(12,30),(13,37),(14,42),(15,47),(16,40),(17,45),(18,38),(19,43),(20,48),(21,41),(22,46),(23,39),(24,44)]])
45 conjugacy classes
| class | 1 | 2A | 2B | 2C | ··· | 2H | 2I | 2J | 3A | 3B | 3C | 4A | 4B | 4C | 4D | 4E | 4F | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 6I | ··· | 6N | 12A | 12B | 12C | ··· | 12I | 12J | 12K |
| order | 1 | 2 | 2 | 2 | ··· | 2 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | ··· | 6 | 12 | 12 | 12 | ··· | 12 | 12 | 12 |
| size | 1 | 1 | 2 | 6 | ··· | 6 | 18 | 18 | 2 | 2 | 4 | 2 | 2 | 6 | 6 | 18 | 18 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 12 | ··· | 12 | 2 | 2 | 4 | ··· | 4 | 12 | 12 |
45 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | S3 | D6 | D6 | D6 | D6 | D6 | D6 | 2+ 1+4 | S32 | C2xS32 | C2xS32 | D4:6D6 | D4oD12 | D12:24D6 |
| kernel | D12:24D6 | D12:5S3 | D12:S3 | S3xD12 | D6:D6 | S3xC3:D4 | C6xD12 | C3xC4oD12 | C12.59D6 | C2xD12 | C4oD12 | Dic6 | C4xS3 | D12 | C3:D4 | C2xC12 | C22xS3 | C32 | C2xC4 | C4 | C22 | C3 | C3 | C1 |
| # reps | 1 | 2 | 2 | 2 | 2 | 4 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 5 | 2 | 2 | 2 | 1 | 1 | 2 | 1 | 2 | 2 | 4 |
Matrix representation of D12:24D6 ►in GL4(F13) generated by
| 5 | 5 | 0 | 0 |
| 8 | 0 | 0 | 0 |
| 0 | 0 | 8 | 8 |
| 0 | 0 | 5 | 0 |
| 0 | 0 | 8 | 8 |
| 0 | 0 | 0 | 5 |
| 5 | 5 | 0 | 0 |
| 0 | 8 | 0 | 0 |
| 0 | 0 | 10 | 3 |
| 0 | 0 | 6 | 3 |
| 6 | 3 | 0 | 0 |
| 10 | 7 | 0 | 0 |
| 6 | 3 | 0 | 0 |
| 10 | 7 | 0 | 0 |
| 0 | 0 | 10 | 3 |
| 0 | 0 | 6 | 3 |
G:=sub<GL(4,GF(13))| [5,8,0,0,5,0,0,0,0,0,8,5,0,0,8,0],[0,0,5,0,0,0,5,8,8,0,0,0,8,5,0,0],[0,0,6,10,0,0,3,7,10,6,0,0,3,3,0,0],[6,10,0,0,3,7,0,0,0,0,10,6,0,0,3,3] >;
D12:24D6 in GAP, Magma, Sage, TeX
D_{12}\rtimes_{24}D_6 % in TeX
G:=Group("D12:24D6"); // GroupNames label
G:=SmallGroup(288,955);
// by ID
G=gap.SmallGroup(288,955);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,219,675,80,1356,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^12=b^2=c^6=d^2=1,b*a*b=c*a*c^-1=a^-1,d*a*d=a^5,c*b*c^-1=a^4*b,d*b*d=a^10*b,d*c*d=c^-1>;
// generators/relations